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BIBLIOGRAPHIC RECORD TARGET
Graduate Library
University of Michigan
Preservation Office
Storage Number:
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UL FMT B RT a BL m T/C DT 07/18/88 R/DT 07/18/88 CC STATmmE/Ll
010: : I a 05028039
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040: :  a MnU  c MnU  d MiU
050/1:0: aQA371 b.F7
100:1 : I a Forsyth, Andrew Russell,  d 18581942.
245:00:  a Theory of differential equations.  c By Andrew Russell Forsyth.
260: :  a Cambridge, [ b University Press, jcl8901906.
300/1: : la4pts. in6v. c23cm.
505/1:1 :  a pt I. (Vol. I) Exact equations and Pfaff s problem. 1890.pt.
11. (Vol. IIlII) Ordinary equations, not linear. 1900.pt. III. (Vol. IV)
Ordinary equations. 1902.pt. IV (vol. VVI) Partial differential equations.
1906.
590/2: : aastr: Pt.l (vol 1) only
650/1:0:  a Differential equations
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THEORY
DIFFERENTIAL EQUATIONS.
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aouOon; 0. J. CLAY and SONS,
CAMBKIDGE UNIVERSITY PEESS WAREHOUSE,
LANE,
NGTON STREET,
m: F. A. EROCKHAUS.
TUE MAOMILLAN COMPANY.
ciilfa: MACMILLAN AND 00., Lru.
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THEOEY
OF
DIFFEEENTIAL EQUATIONS.
PAUT III.
OKDINAKY LINEAR EQOATIONS.
ANDREW EUSSELL FOBSYTH,
ScD., LL.D., F.K.S.,
CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1902
All rights r^ssrued.
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PBEPACE.
The present volume, constituting Part III of this
work, deals with the theory of ordinary linear differential
equations. The whole range of that theory is too vast to
be covered by a single volume ; and it contains several
distinct regions that have no organic relation with one
another. Accordingly, I have limited the discussion
to the single region specially occupied by applications
of the theory of functions ; in imposing this limitation,
my wish has been to secure a uniform presentation of
the subject.
As a natural consequence, much is omitted that
would have been included, had my decision permitted
the devotion of greater space to the subject. Thus the
formal theory, in its various shapes, is not expounded,
save as to a few topics that arise incidentally in the
functional theory. The association with homogeneous
forms is indicated only slightly. The discussion of com
binations of the coefficients, which are invariautive under
all transformations that leave the equation linear, of the
associated equations that are covaviantive under these
transformations, and of the significance of these invariants
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and covariants, is completely omitted. Nor is any appli
cation of the theory of groups, save in a single functional
investigation, given here. The student, who wishes to
consider these subjects, and others that have been passed
by, will find them in Schlesinger's Handhuch der Theorie
der linearen Differentialgleichungen, in treatises such as
Pieard's Corns d'Analyse, and in many of the memoirs
quoted in the present volume.
In preparing the volume, I have derived assistance
from the two works just mentioned, as well as from the
uncompleted work by the late Dr Thomas Craig. But,
as will be seen from the references in the text, my main
assistance has been drawn from the numerous memoirs
contributed to learned journals by various pioneers in the
gradual development of the subject.
Within the limitations that have been imposed, it
will be seen that much the greater part of the volume is
assigned to the theory of equations which have uniform
coefficients. When coefficients are not uniform, the
difficulties in the discussion are grave : the principal
characteristics of the integrals of such an equation have,
as yet, received only slight elucidation. On this score,
it will be sufficient to mention equations having algebraic
coefficients : nearly all the characteristic results that have
been obtained are of the nature of existencetheorems,
and little progress in the difficult task of constructing
explicit results has been made.
Moreover, I have dealt mainly witli the general
theory and have abstained from developing detailed
properties of the functions defined by important par
ticular equations. The latter have been used as illustra
tions ; had they been developed in fuller detail than is
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given, the investigations would soon have merged into
discussions of the properties of special ■ functions. In
stances of such transition are provided in the functions,
defined by the hypergeometric equation and by the
modern form of Lamp's equation respectively.
A brief summary of the contents will indicate the
actual range of the volume, In the first Chapter, the
synectic integrals of a linear equation, and the conditions
of their uniqueness, are investigated. The second Chapter
discusses the general character of a complete system of
integrals near a singularity of the equation. Chapters
III, IV, and V are concerned with equations, which have
their integrals of the type called regular ; in particular.
Chapter V contains those equations the integrals of which
are algebraic functions of the variable. In Chapter VI,
equations are considered which have only some of their
integrals of the I'egular type ; the influence of such
integrals upon the reducibility of their equation is in
dicated. Chapter VII is occupied with the determination
of integrals which, whUe not regular, are irregular of
specified types called normal and subnormal ; the
functional significance of such integrals is established,
in connection with Poincare's development of Laplace's
solution in the form of a definite integral. Chapter VIII
is devoted to equations, the integrals of which do not
belong to any of the preceding types ; the method of
converging infinite determinants is used to obtain the
complete solution for any such equation. Chapter IX
relates to those equations, the coefficients of which are
uniform periodic functions of the variable : there are two
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classes, according as the periodicity is simple or double.
The final Chapter deals with equations having algebraic
coefficients; it contains a brief genera! sketch of Poincare's
association of such equations with automorphic functions.
In the revision of the proofsheets, 1 have received
valuable assistance from three of my friends and former
pupils, Mr. E. T. Whittaker,M.A., and Mr. E. W. Barnes,
M.A., Fellows of Trinity College, Cambridge, and Mr,
E. W. H. T. Hudson, M.A., Fellow of St John's College,
Cambridge ; I grateftilly acknowledge the help which
they have given me.
And I cannot omit the expression of my thanks to the
Staff of the University Press, for the unfailing courtesy
and readiness with which they have lightened my task
during the printing of the volume.
A. R. FORSYTH.
Trisity Cult.bge, Cambeidgb,
1 jtfareA, 1902.
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CONTENTS.
CHAPTER I.
LINEAR EQUATIONS : EXISTENCE OF SYNECTIC INTEGRALS ;
FUNDAMENTAL SYSTEMS.
Introductory remarks
Form of the homogeneous linear equation of oidei m
Eatablisbmeut of the esistence of a sjneutii, integrril in the
domeiin of an ordinary puint, dotennined uniquely by the
initial conditions with corollanea, and Bsamples
Hermite's treatment of the equation with constant tooflicienta
Continuation of the aynectie integral beyond the initial
domain ; r^on of its continuity bounded by the singu
larities of the equation
Certain deformations of path of independent variable leave
the final int^ral unchanged ......
Seta of integrals determined by sets of initial valuea .
The determinant ^ {z) as affecting the linear independence
of a sot of m integials a fundamental system and the
eSective test of it* fitness
Any integial is linearly e:£press!ble in terms of the elements
of a fundlmentul sjstem .....
Construction if i speiiil fundamental systeni
CHAPTER II.
GENERAL FORH AND PROPERTIES OP INTEGRALS NEAR A SINGULARITY.
13. Constructitn of the fundwmentcd equation belonging to a
singularity 35
14. The fundimental equation is independent of the choice of the
f\indimcntal aytera Poincarii'a theorem ... 38
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CONTENTS
Tl 1 m tay d &ors f tt f d t ! q
it la tal f m L n
T y 13 p joat w b 11
4. fi d m ta] y t f t^r 1 1: th
f dm tal ©q tat diat t f
Eff t f m Itpl oot
T fmt fthfdmtleqt
1 f th d ced f n
l_ 1 f teiral ted with Itil
fth g p t bgi [
TI bmg b g 1 eq t OB h t
tegral t g" P d th g
i t th fegnl
■y I hoat f th Ijt 1 I f th
Hfti b bg p
ult th teg al
1
IM
f 1
relating to such integrals .
f tl
lyt I
CHAPTER III.
RRQULAR ISTEGBALS 1 EQUATION HAVING ALL ITS INTEGRALS RBRTJI.AI
NEAR A SINGULARITY.
39. Definition of integral, regular in the vicinity of a singularity :
index to which it belongs 7
30. Iddes of the dctemiioant of a fundamental system of integrals
all of which are regular near the singularity ... 7
31. Form of homogeneous linear equation when all its integrals
are regular near a 7
32. 33. Converse of the preceding reault, established by the method
of Frobenius ......... 7
34. Series proved to converge uniformly ajid unconditionally . 8
35, Integral associated with a simple root of an algebraic (in
dicia!) equation ..,.■... i
36—38. Sot of int^rals associated with special group of roots of the
algebraic (indicial) equation, with summaj'y of results
when all the integrals are regular E
39. Definition of indieial equation, indidol funetiiyii : signiiicance
of integrals obtained i
40. The iiit^rals obtained constitute a fundamental system ; with
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Conditiona that ever3 iPgular integral belonging to a par
ticulir exiionent '.hould have its espression free from
logiiithms , with es^mple'J ......
Condititni that there should be at least one r^ular integral
belonging to a partitulai exponent and free from loga
iithms ......
Alternitue method aometimea effective for settling the ques
tion in §§ 43, 43
Discnmination between real singularity and apparent singu
lant; conditions fot the latter .....
Noh on thp leties in 1) S4
CHAPTER IV,
EQUATIONS HAVING THEIR INTEGKALS REGULAR IN THE VICINITY OF
EVERY SINGULARITY (iKCLOUING infinity).
46. Equations (said to be of the FacAsian type) having all their
int^r&ls r^ular in the vicinity of every singularity
{including =o ) ; their form : with examples . . . 123
47. Equation of second order completely determined by assign
ment of singularities and tlieir exponents : Riemaun's
jpfunction 135
48. Significance of the relation among the exponents of the pre
ceding equation and fimotion 139
49. Oonatruction of the differential equations thus determined . 141
50. The equation satisfied by the hype^eometric series, with
51. 52. Equations of the Fuclisian type and the second order with
more than three singularities (i) when co is not a singu
larity, (ii) when oo is a singularity ..... 150
53. Normal forme of such equations 156
54. Lamp's equation transformed so as to be of Fuchsiaa type . 160
55. ■ B6clier'a theorem on the relation between the linear equations
of mathematical physics and an equation of the second
order and Fuchaian type with five singularities . . 161
56. Heine's equations of the second order having an integral that
is a polynomial 165
57. Equations of the second order all whose integrals are rational 169
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CHAPTER V.
1 KQUATIONS OP THE SECOND AKD THE THIRD OEDERS
i ALGEBRAIC I
Mutlnj la of letermining whether an equation hi^ dlt,pl i ii
ntegi ill
Kle a a special meth id for detonnining all the finite groups fc
thp equation of the ^uond ordei
Ihp pq ationa satisfied by the quotient of two 'olutionj
the equition of the second order then integrals
L m'itiurtion of equations of the second order a"
ntegrable
Meini of deteiminmg whether a gi\eii equation is n
braicaily uitegralle with examples
Lquitims jf the third order their quotient eq lat ons
Painleve >■ invariants, cirreaponding to the Scliwaiziin der
tive for the equation of the second order connet
with La^uerres mvanant
A'.i loiation with finite group? of trai aformat oi
1 neolinear in two variables
IndiLitims Df other poMible methods
EemiikJ on equatmn^ of the foiuth. orler
Association of equations of the third ani highei triers
the theon of homogeneous forms
And of equati >ns of the second order
"Diacuasion of equations of the thiri order with i "e
theorem due to Fuchi with cs'implp in I roferei <
e j^uations of higher order
th^t '
CHAPTER VI.
lUA'l'IONS HAVING ONLY SOME OF THEIR INTEGHALS REGULAR
NEAR A SINGULARITY.
Equations having only some of their integrals regular ii
vicinity of a singularity : the characteristic indes .
The linearly independent aggregate of r^ular integrals
satisfy a liiieaj equation of order equal to their number ,
Reducible equations
Frobenius's characteristic function, indicial function, ivdiciid
eqwitwji ; normal form of a differential equation associ
able with the indicia] function, uniquely determined b;
the characteristic function
The number of r^ular iutegrals of an equation of order ii
and oharaoterietic index n is not greater than m — n
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CONTENTS
Tlr mbe f gul tp ! be 1 sa th
D te t f tl regul t 1 h tii y t
w th mpl
E t f irred bl equat
Aqt fil h g dpedt ulor te
b las grl t 1 a. tedw th
it f rd
La eqtndf to^n qt
Eelt htee qt dtadit, pett f
th L fheil Ipdtrel t orals
Jia dhlt I t
CHAPTER Vn.
NORMAL integrals: suenosmal integrals.
Integrals for which the singularity of the equation is
essential : normal int^rals
Thome's method of obtaining norma! integrals when they
Construction of determining factor : possible cases .
Svhnormal integrals
Rank of an equation ; Poincar^'s theorem od a set of
normal or subnormal functions as integrals ; examples
Hftmbu:^er's equations, having s=0 for an essential singu
larity of the integrale, which are regular at tc and
elsewhere are sjnectic : equation of second order
Cayley's method of obtaining normal iut^rals .
Hamburger's equations of order higher than the second
Conditions associated with a simple root of the character
istic equation for the determining factor .
Likewise for a multiple root
Subnormal int^rals of Hamburger's equations
Detailed discussion of equation of the third order .
Normal integrals of equations with rational coefficients
Poincar^s development of Laplace's solution for grad(
1. Liapouuoff's theorem .......
1—105. Application to the evaluation of the definite int^ral ir
Laplace's solution, leading to a normal integral
i. Doubleloop integrals, after Jordan and Pochhammer
'. When the normal series diverges, it is an asymptotic repre
sentation of the definiteintegral solution
i. PoincarS's transformation of equations of rank higher than
unity to equations of rank unity
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CHAPTER V1.IT.
INFINITE DETERMISANTS, AND THEIK APPLICATION TO THE SOLUTION
OF LINBAU EQUATIONS.
109. lutljluU 311 Ji iniimte determnints tLsts of c l\OY
s;cnci, 1 io[>eities . 348
110. Dciebitiicnt . 352
111. Mm I'l 353
112. IT[ ftrm comergence when t iistitui.nt< aie fun t ons of ii.
paiametei . 358
113. Solution of an unlimited r umbct of simultaneous In ear
equations . 360
114. Difletential equations having no regular totegial oo noimal
integial no subnormal integtal 363
115. Infegral in the form (f a Laurent aenes intioducticn of
!iii inhmte determinant ii{pj ■ SGii
IIG. C nvergent* ot Q(p) . 366
117. Introiuction of i^nother inimite deteiminint I>{p) its
convergente and its telation to Q{p) with deduced
eiiieSBion of £2(p) . 369
118. ( niei^ence of tlie Liurent ^nea exj reding the mtegial . 376
119. & neiihsation of mpthod ol Fiobennis (in Ohap Hi) to
determine a system cf mtBgrah . 379
120 — 123. 'V annua cases atoordmg t) the chaiatter of the ineduLiblc
r ots of i>{p) = . 380
124. The i^stem ot integrals i5 fundamental . 387
126. Th equati n l>{p)=0 is efiectneh the fundaineatal equa
tion for the combiiidtnn of Bingiilantiei within the
cu'cle I =fi . 389
126. (.eneiil remark example'. . 392
127. Other methc h of ibtaining the fundamental equati ii to
which D (fi)=0 IS efiectively equivalent with an
example . 398
CHAPTER IX.
EQUATIONS WITH UNIFOHII PLRIODIC f DLIf ICIENTS.
Equationfl with mmply 'penodu, coefhcients the funda
mental er[uation a sooiated with the juniod
Siiiple looti f the fund^mentil equat on
\. iQultii le 1 not of the fundamental equiti ti .
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131. Analytical form of tte integrals associated with a root . 411
132. Modification of the form of the group of int^rals associated
with a multiple root 414
133. Use of elementary divisors : resolution of group into sub
groups : numbei' of integrals, that are periodic of the
second kind 416
134. More precise establiahment of results in § 133 . . . 417
135. Converse proposition, analogous to Fuchs's theorem in § 25 420
136. Further determination of the integrals, with examples . 421
137. Liapounoflfs method 425
138 — 140. Discussion of the equation of the elhptic cylinder,
v/' + {aiocOB^)w=^0 . . . .431
141. Equations with doublyperiodic coefficients; the funda
mental equations associated with the periods . . 441
143, 143. Picard's theorem that such an equation poasessea an
int^ral which is doubly periodic of the second kind :
the number of such integrals 447
144, 145. The int^rals associated with multiple roots of the funda
mental equations ; two cases ..... 451
146. First stage in the construction of analytical expressions of
integrals ......... 457
147. Equations that have uniform integrals : with examples . 459
148. Lamp's equation, in the form vf=w{n{n + \)i^{z)^B),
deduced from the equation for the potential . . 464
149 — 151. Two modes of constructing the integral of Lame's equation 4li8
CHAPTER X.
EQUATIONS HAVING ALGEBRAIC COEFFICIENTS.
152. Equations with algebraic coefficients 478
153, 154. Fundamental equation for a singularity, and fundamental
systems; examples 480
155, 156. Introduction of automorphic functions .... 498
167, 158. Automorphic property and oonformal representation . 491
159—161. Automorphic property and linear equations of second order 495
162. Illustration from elliptic functions 501
163. Equations with one singularity 506
184. Equations with two singularities 507
165. Equations with three singularities 508
166. General statement as to equations with any number of
singularities, whether real or complex . . . BIO
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statement of Poiucar^'s results
Poincar^'s theorem that any linear equation with algebraic
coefficients can be integrated by Fucbsian and Zeta
fucheian functions
Properties of these functions : and verification of Poincar^'s
theorem
Concluding remarks
IN BBS I
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CHAPTER I.
Linear Equations ; Existence of Synectic Integrals :
Fundamental Systems.
1. The course of the preceding investigations has made it
manifest that the discussion of the properties of functions, which
are defined by ordinary differential equations of a general t3rpe,
rapidly increases in difficulty with successive increase in the order
of the equations. Indeed, a stage is soon reached where the
generality of form permits the deduction of no more than the
simplest properties of the functions. Special forms of equations
can be subjected to special treatment ; but, when such special
forms conserve any element of generality, complexity and difficulty
arise for equations of any but the lowest orders. There is one
exception to this broad statement ; it is constituted by ordinary
equations which are hnear in form. They can be treated, if not
in complete generality, yet with sufficient fulness to justify their
separate discussion ; and accordingly, the various important results
relating to the theory of ordinary linear differential equations
constitute the subjectmatter of the present Part of this Treatise.
Some classes of linear equations have received substantial
consideration in the construction of the customary practical
methods used in finding solutions. One particular class is com
posed of those equations which have constants as the coefficients
of the dependent variable and its derivatives. There are, further,
equations associated with particular names, such as Legendre,
Bessel, Lam^ ; there are special equations, such as those of the
hypergeometric series and of the quarterperiod in the Jacobian
theory of elliptic functions. The formal solutions of such equations
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2 HOMOGENEOUS [1 .
can be regarded as known; but so long as the investigation is
restricted to the practical construction of the respective series
adopted for the solutions, no indication of the range, over which
the deduced solution is valid, is thereby given. Ifc is the aim of
the general theory, as applied to such equations, to reconstruct
the various methods of proceeding to a solution, and to shew
why the isolated rules, that seem so sourceless in practice, actually
prove effective. In prosecuting this aim, it will be necessary to
revise for linear equations all the customarily accepted results, so
as to indicate their foundation, their range of validity, and their
signiiicance.
For the most part, the equations considered will be kept as
general as possible within the character assigned to them. But
from titne to time, equations will be discussed, the functions
defined by which can be expressed in terms of functions already
known ; such instances, however, being used chiefly as illustrations.
For all equations, it will be necessary to consider the same set of
problems as present themselves for consideration in the discussion
of unrestricted ordinary equations of the lowest orders : the exist
ence of an integral, its uniqueness as determined by assigned
conditions, its range of existence, its singularities (as regards
position and nature), its behaviour in the vicinity of any singu
larity, and so on : together with the converse investigation of the
limitations to be imposed upon the form of the equation in order to
secure that functions of specified classes or types may be solutions.
As is usual in discussions of this kind, the variables and the
parameters will he assumed to be complex. It is true that, for
many of the simpler applications to mechanics and physics, the
variables and the parametei'S are purely real ; but this is not the
case with all such applications, and instances occur in which the
characteristic equations possess imaginary or complex parameteis
or variables. Quite independently of thk latter fact, however, it
is desirable to use complex variables in order to exhibit the proper
ielation of functional variation.
2. Let z denote the independent variable, and w the dependent
variable ; z and w varying each in its own plane. The differential
equation is considered linear, when it contains no term of order
higher than the first in w and its derivatives ; and a linear equation
is called homogeneous, when it contains no term independent of w
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2.] LINEAR EQUATIONS 3
and its derivatives. By a wellknown formal result*, the solution
of an equation that is not homogeneous can be deduced, merely by
quadratures, from the solution of the equation rendered homo
geneous by the omission of the term independent of w and its
derivatives ; .and therefore it is sufficient, for the purposes of the
general investigation, to discuss homogeneous linear equations.
The coefficients may be uniform functions of s. either rational or
transcendental ; or they may be multiform functions of a, the
simplest instance being that in which they are of a form (s, z),
where is rational in s and z, and s is an algebraic function of z.
Examples of each of these classes will be considered in turn. The
coefficients will have singularities and (it may be) critical points ;
all of these are determinable for a given equation by inspection,
being fixed points which are not affected by any constants that
may arise in the integration. Such points will be found to include
all the singularities and the critical points of the integrals of the
equation ; in consequence, they are frequently called the singu
larities of the equation. Accordingly, the differential equation,
assumed to be of order m, can be taken in the form
where the coefficients p^, p^, ..., pm are functions of s. In the
earlier investigations, and until explicit statement to the contrary
is made, it will be assumed that these functions of z are uniform
within the domain considered ; that then' singularities are isolated
points, so that any finite part of the plane contains only a limited
number of them : and that all these singularities (if any) for finite
values of s are poles of the coefficients, so that their only essential
singularity (if any) must be at infinity. Let f denote any point in
the plane which is ordinary for all the coefficients p ; and let a
domain of ^ be constructed by taking all the points z in the
plane, such that
2fi«i«fi.
where a is the nearest to ^ amohg all the singularities of all the
coefficients. Then within this domain (but not on its boundary)
we have
P.P.i't). (»1.2 »).
* See mj Treathte on Differential Eqnatiinui, § 75.
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4 SYNECTIC [2.
where P„ denotes a regular function o{ s — ^, which generajly is
an infinite series of powers of z — f converging within the domain
of ^. An integral of the equation existing in this domain is
uniquely settled hy the following theorem ; —
In the domain of an ordinary point ^, the differential equation
possesses an integral, which is a regular function of z — ^ and, with
its first m — 1 derivatives, acquires arbitrarily assigned values when
z = Z; and this integral is the only regular function of z—^ in
the specified domain, which satisfies the equation and fulfils the
assigned conditions*.
The integral thus obtained will be calledf the synectio integral.
Synectic Integrals.
§. The existence of an integral which is a holomorphic
function of s— ^ within the domain will first be established.
Let r he the radius of the domain of i^; let M,, ..., M^ denote
quantities not less than the maximum values of \p,], ..., \pm\
respectively, for points within the domain ; and let dominant
ictions ^, ...
, <ini, defined by the expressions
constructed.
Then*
for every positive integer a. The dominant functions ^ are used
to construct a dominant equation
^ = ■ft S^S=r + <^ rf^S^ +  + ■^'"«.
which is considered concurrently with the given equation,
* The conditions, as to the arbitrarily assigned values to be ftotiuired at f by tu
and itfl derivatives, are called the initial conditions ; the values are called the
initial values,
t As it is a regular function of the variable, it would have been proper to call
it the regular int^cal. This term has however been appropriated [sec Chapter iii,
§ 39) to describe another class of integrals of linear equations; as the use in this
other conneciion is now widespread, oonfusion would result if the use were changed.
J See mj Theory of Functions, 2aA edn,, §22; quoted hereafter as ?'. J''.
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3,] INTEGRALS 5
Any function which is regular in the domain of ^ can be
expressed as a converging series of powers of ^~f; and the
coefficients, save as to numerical factors, are the values of the
various derivatives of the function at f Accordingly, if there is
an integral w which is a regular function of 5 — J^, it can be formed
when the values of all the derivatives of w at £f are known. To
w. r , ., ~, , the arbitrary values specified in the initial
conditions are assigned. All the succeeding derivatives of w can
be deduced from the differential equation in the form
" rf^"""
I A^,^',
(for a — m, m + 1, ... ad inf.), by processes of differentiation,
addition, and multiplication: as the coefficient of the highest
derivative of w in the equation (and in every equation deduced
from it by differentiation) is unity, new critical points are not
introduced by these processes, so that all the coefficients A are
regular within the domain of f.
The successive derivatives of u are similarly expressible in the
(for a = m, m + 1, ... ad inf.), obtained in the same way as the
equation for the derivatives of w. The coefficients B have the
same form as the coefficients A, and can be deduced from them by
changing the quantities p and their derivatives into the quantities
tp and their derivatives respectively.
The values of the derivatives of w and u a,t ^ are required.
When i^ = J", all the terms in each quantity B are positive ; on
account of the relation between the derivatives of the quantities p
and <b, it follows that
B«>^«, (s = l, ...,m),
.... , „, , \dw\ I d'^'^w I ,
when 2 = f. Let the mitial values oi wl, j . ■■, . m^' > when
z = ^, be assigned as the values of u, f ,■■■• j^i^i when 2 = ^;
then
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6
EXISTENCE
OF
when«
r.
for the valu
Mm, m + 1,
... oi
:». If the
, series
(») + (2
/A.\ (i
0
"Ui +
2!
converge!
"©
denotes the
v.,.e„f^'.
i'hen z
series
where ( r— 1 denotes the value of ^— when z=t, also converses ;
Vets"/ ffla" °
it then represents a regular function oi z~ ^ which, after the mode
of formation of its coefficients, satisfies the differential equation.
We therefore proceed to consider the convergence of the series
for u, obtained as a purely formal solution of the dominant equa
tion. To obtain explicit expressions for the various coefficients in
this series, let s — f = rw, taking x as the new independent variable.
Points within the domain of f are given by a;< 1 ; and the
dominant equation becomes
ax™ s=i dx"^'
When the .series for u, taken in the form
is substituted in the equation which then becomes an identity, a
comparLsoQ of the coefficients of a^ on the two sides leads to the
relation
holding for all positive integer values of k.
This relation shews that all the coefficients h are expressible
linearly and homogeneously in terms of 6o, 6i, ..., hwi and that, as
the first m of these coefficients have been made equal to the moduli
of the in arbitrary quantities in the initial conditions for w and
therefore are positive, all the coefficients h are positive. Hence
k + M,r,
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S.] A SYNECTIC INTEGRAL 7
By the initial definition of M,, it was taken to be not less than
the maximum value of \pi\ within the domain of f; it can there
fore be chosen so as to secure that M^r > m. Assuming this
choice made, we then have
OjK+l > Om+i;— 1 >
so that the successive coefficients
From the difference equation satisfied by the coefficients b, it
follows that
V+*. k + ni ,^2 (m + k)l ' t™+ft_,'
So far as regards the m — 1 terms in the summation, the ratio
^m+iis ^ i'm+fti is less than unity for each of them ; Mgi^ is finite
for each of them; and \in + k — s)\^{m+k)\ is zero for each of
them, in the limit when k is marie infinite. Hence we have
and iherefoi'e
<1.
for points within the domain of f, so that* the series
converges within the domain of f. The convergence is not estab
lished for the boundary, so that it can be affirmed only for points
within the domain; it holds for all arbitrary positive values
assigned to b^, b^, ..., b^i.
It therefore follows that, at all points within the domain of ^,
a regular function of s — i^ exists which satisfies the original
differential equation for tv, and, with its first m — 1 derivatives,
acquires at f arbitrarily assigned values.
4. Now that the existence of a synectic integral is established,
the explicit expression of the integral in the form of a powerseries
in z — ^, this series being known to converge, can be obtained
" Cbrystal's Algelra, vol. ii, p. 121.
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8 UNIQUENESS OF [4.
directly from tlie equation. As f is an ordinary point for each of
the coefficients p, we have
p,^PA.^0, (s = l, 2, ...,m),
where Pg denotes a regular function of a — f. Let a^, oli, ..., ow,
be the arbitrary values assigned to w, j— , ..., j ^~ , when s = f ;
and take
which manifestly satisfies the initial conditions. In order that
this may satisfy the equation, it must make the equation an
identity when the expression is substituted therein. When the
substitution is effected, and the coefficients of {e — ^y on the two
sides of the identity are equated, we have a relation of the form
where A^+s is a linear homogeneous function of the coefficients a,,
such that K<m + s, and is also linear in the coefficients in the
quantities P, (3— f), ..., P„(2 — f); and the relation is valid for
s= 0, 1, 2, ..., ad inf. Using the relation for these values of s in
succession, we find a^, a^+i, Om+a ■■■ expressed (in each instance,
after substitution of the values of the coeiScients which belong to
earlier values of s) as a linear homogeneous function of the quanti
ties Oj, «!, ..., ctntii and in am+s, the expressions, of which the
initial constants a^, a^, ..., 0^1 are coefficients, are polynomials of
degree s + 1 in the coefficients of the functions P,{s — ^), ....
Pm (^  ^) The earlier investigation shews that the powerseries
for w converges ; accordingly, the determination of the coefficients
a in this manner leads to the formal expression of an integral w
satisfying the equation.
5, Further, the integral thus obtained is the only regular
function, which is a solution of the equation and satisfies the
initial conditions associated with a^, eij, ..., 0^1. If it were
possible to have any other regular function, which also is a solu
tion and satisfies the same initial conditions, its expression would
be of the form
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5.] THE STNECTIC INTEGRAL 9
a regular function oi z — i^. The coefficients would be determin
able, as before, fi'om a relation
whore ^'m+j is the same function of a^, ..., a^i, «'m. ■■, <t'ni+ai
as ^m+s is of Oo, ..., a^i, «,„, ..., am+E_i Hence
a'jii+i = ^'in+1 = dm+i> after substitution for a'^,
and so on, in succession. The coefficients agree, and the two
series are the same, so that w = w' ', and therefore the initial con
ditions uniquely determine an integral of the equation, which is a
regular function of ^ — f in the domain of the ordinary point ?.
Corollary I. If all the initial constants ag, a,, ..., a^i are
zero, then the synectic integral of the equation is identically zero.
For in the preceding discussion it has been proved that om+e, for
all the values of s, is a linear homogeneous function of a^, ...,
Ooti ; hence, in the circumstances contemplated, a,n+s = for all
the values of s. Thus every coefficient in the series vanishes ;
accordingly, the integral is an identical zeio.
CoEOLLARY II. The initial constants a^, a^, ..., am_i occur
linearly in the ea^ession of the synectic integral ; and each of the
m variable quantities, which have those constants /or coe^cients, is
a synectic integral of the equation. The first part is evident,
because all the coefficients in w are linear and homogeneous in
OoiO:,, ..., cWi. As regards the second part, the variable quantity
multiplied by Sg is derivable fi'om w by making a^ = 1, and all the
other constants a equal to zero ; these constitute a particular set
of initial values which, according to the theorem, determine a
synectic integral of the equation. Thus the synectic integral,
determined by the initial values a^, ..., "mi, is of the form
aoUi + aiU^+ ... iam_iJfm,
where each of the quantities m,, u^, ...,«„ is a synectic integral of
the equation.
j!fote 1. The series of powers oi z — ^, which represents the
synectic integral, has been proved to converge within the domain
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10 EXISTENCE OF [5.
of Z, SO that its radius of convergence is  «  f  , where a is the
singularity of the coefficients which is nearest to f. All these
singularities lying in the 6nibe part of the plane are determinable
by mere inspection of the forms of the coefficients : another
method must be adopted in order to take account of a possible
singularity when z = co because, even though a = oo may be aji
ordinary point of the coefficients, infinite values of the variable
affect the character of w and its derivatives.
For this purpose, we may change the variable by the substi
tution
ea:= 1,
and we then consider the relation of the a'origin to the trans
formed equation as a possible singularity. The transformation of
the equation is immediately obtained by means of the formula
#w , ,,t * fc ! (fe  1) ! a^+° d'w^
dz" ^ ^ ,Z^aL\{a.\)\{ka)\ dx'
inspection of the transformed equation then shews whether x = ()
is, or is not, a singularity. Or, without changing the independent
variable, we may consider a series for w in descending powers of z :
pies will occur hereafter.
It may happen that there is no singularity of the coefficients
in the finite part of the plane, infinite values then providing the
only singularity. In that case, we should not take the quantity r
in the preceding investigation as equal to [co — ^], that is, as
infinite ; it would suffice that r should be finite, though as large
as we please.
It may happen that there is no singularity of the coefficients
for either finite or infinite values of s; if the coefficients are
uniform, they then can only be constants. The dominant equa
tion is then effectively the same as the original equation ; the
investigation is still applicable, but it furnishes less information
as to the result than a method which will be indicated later (§ 6).
Note 2. The preceding proof is based upon that which is
given* by Fuchs in his initial, and now classical, memoir on the
theory of linear differential equations.
* Creile, t. lkvi (1866), pp. 133—135.
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5.] A SYNECTIC INTEGRAL 11
The theorem can also be established by regarding it as a
particular case of Cauchy's theorem, which relates to the posses
sion of unique synectic integrals by a system of simultaneous
equations. If
_ ii"w
™'' ~ ~3^ '
the homogeneous linear equation of order
the system
i = Wj+i, for s = 0, 1, ....
(is
These equations possess integrals, expressible as regular functions
of if—?, such that w„, Wi, ...,w™_, assume arbitrarily assigned
values when e — ^, and the integrals are unique when thus
determined : which, in effect, is the theorem as to the syuectic
integral of the hnear equation*.
Note 3. A different method for establishing the existence of
the integrals, though if. does not indicate fully the region of
(a =
^0, 1
»I),
1 be
replaced by
m^2,
ipmW<
their convergence, can be based upon
Giintherf. It consists in the adoption
a suggestion
of another i
made by
subsidiary
equation
where
... +fm'V,
*'{^'^r
for^=l, ...
its integrals
are
The advantage of this form of equati<
explicitly given in the form
an is that
„.f, i^fr
where a is a root of the equation
(7 (t  1) ... (<7 m + 1) =  rjlf,^ (o  1) ... ((7 ^ m + 2)
+ ^M,a (<7  1) . . . (ff  m + 3) + . . .
+ { l)™'r"'W™_,.7 + ( l)'"r™M^.
* See Part ii of this Treatise, gg 4, 10—13.
I CreUe, t. csviii (1897J, pp. Sol— 3.13 ; see also some remarks thereupon b;
Fuohs, a.,pp. 354, 353.
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12 EXAMPLES [5.
If a root a is multiple, the corresponding group of integrals is
easily obtained*.
The construction of the actual proof on the foregoing lines is
left aa an exercise.
Ex. 1. Consider the equation
h arity of the coeffl
1 d n P y to the immediate
po he coefficients of the
a, idi unity. The equa
gr wh h ries of powers of z
q y d te mm d hy the conditions
d & wy constant.^. To
which then must be an identity. In order that the coefficient of 2" may
vanish after substitution, we must have
(« + a)(m + I)6„^.5{«2 + «^)6„=0,
Now hy the initial conditions, we have
h^ = a, 6i = 3;
hence
" See my Treatine on Differential Equations, %% 47, 43.
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5.] EXAMPLES
and, similarly,
i products baing taken for integer values of s from 1 to m. The
syaectic integral satisfying the initial conditions is
both series, if infinite, converging for values of i such that  ^  < 1
The best known instance of this equation is that whict is usually asso
ciated with Legendre's name : k then isp(p + l), ondp (in the simplest form)
is a positive integer. If p be an even integer, all the coefBcients b^„, for
2m > p, vanish, so that the quantity multiplying a is then a jwlynomial ; the
quantity multiplying j3 is an infinite series. If p be an odd integer, all the
coefBcients fiam + u f"r 2ni.+ l >p, vanish, so that the quantity multiplying 3
is then a polynomial ; the quantity multiplying a ia an infinite aeries. In all
other cases, the quantitiea multiplyii^ a and (3 are, each of them, infinite
aeries ; in every instance, the aeries converge when  z  < 1.
£V. 2. Obtain the syncctic integral of the equation
(which includes Bessel'a equation as a special case), with the initial conditions
that w = a, T =0 when 2=c, where \o\ > 0.
Ex. 3. Determine the synectic integral of the equation of the hyper
geometric series
"(l)^ + {r(.+»+l).)s*— »,
the initial conditions being that w = A, j =B, when i=^.
E». 4. Determine the synectic integraJa in the domain of £=0,
by the equation
with the initial conditions (i) that w=l, rT=0, when z=0 ;
(ii) that w = 0, £=h wben 2 = 0.
£x. 5. Prove that the synectic integral in the domain of 3 = 0,
by the equation
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14 EQUATIONS WITH [5.
with the initial conditions that vi=l, j = 0, when ^ = 0, is
and if the term in w iuvolving s" he — j s", then
c^, = (,»2 + (2»^« + l)»"Ha3»«(»4)2=^™i «'= + ....
Prove also that the primitive can he espressed in terms of Bessel's functions
of order zero and argument — «="' .
&.. 6. The equation with constant coefficients may he taken in the form
which converges everywhere in the finite part of the piano : and «„, ..., (fmi,
are the arbitrarily assigned initial constants.
Substituting in the diHerential equation this value of w, and equating
coefficients of — 2", wo have
The expression of the coefficients a„, Om+i, ■■■ in terms of «(,, «!, ,.., "mi
depends (by the solution of the foregoing differenceequation) upon the
algebraical equation
"When the roots of ^(e)=0 are different from one another, let them be
denoted by a„ a^, —, a^; and in connection with the m arbitrary constants
flg, tfi, ..., am_i, determine m new constants ^j, A^, ..., A^, by the relations
The determination is unique ; for on solving these m. relations as m, linear
equations in A^, ..., A^, the determinant of the righthand sides is
which is equal to the product of the differences of the roots and is therefore
not zero. Hence, as the constants a^, c,, .... a„_i ore arbitrary, the m new
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5.] CONSTANT COEFFICIENTS 15
constants A^, ..., A^, when iised to replace the former set, can be regarded
aa m independent arbitrary constants. With these constants thus determined,
= £1 2 on'" + '''^K + <'z 2 a^"'*''^jl^+ ... +c^ S o^'^A^,
for all values o! n. When n = 0, we have
2 Ofi"'^^ = CiO„_ + Caa,„,5+ ... +c,„ag = a„^;
when n= 1, wo have
and so on, the general result being that
for all values of n. Hence
= 2 (J.a,' + ^s''/+ ■■■ +^^<%*)^
the customary form of the solution, A^, ..., A^ being m independent arbitrary
constants.
Ex. 7. Apply the preceding method to obtain a similar expression in
finite terms, when the roots of the equation ij>{6)=0 are not all different from
one anotiser.
6. A different method of discussing the linear equation with
constant coefficients has been given by Hermite.
Taking the equation, as before, in the form
we associate with it the expression
(^) = r"  (Ci?™^ + c.r™^^ + . . . + c^).
Denoting \)y f{K) smy polynomial in l^, let
integration being taken round any simple contour in the i^plane.
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16 hermitb's method for equations [6.
In the first place, the degree of the polynomial /(5') may be
taken to be less than m. If initially it is not so, then we have
on division, g (^) being a polynomial, and /, {^) a polynomial of
order less than that of 0, that is, less than m. Now
<!<<,(0<i?o,
round any simple contour in the i^plane ; in the remaining inte
gral, the polynomial is of the form indicated. Accordingly, f(X)
will be assumed to be of order less than m.
We have
taken round the same contour ;
because /(f) is a polynomial and the integral is taken round a
simple contour in the J;'plane. Thus TT is a solution of the
equation.
The only restriction upon f{^) is that, effectively, its degree
must be less than m. It may therefore be taken as the most
general polynomial of degree m — 1 ; in this form, it will contain
m disposable coefficients which can be used to satisfy the initial
conditions. Let these conditions require that, when x = 0, the
variable w and its first m — \ derivatives acquire values ko, hi, ....
km~i respectively ; then we determine /(i^) as follows. Since
we shall dmw the simple contour in the fplane so as to enclose
the origin ; and then the preceding relation shews that, when
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WITH CONSTANT COEFFICIENTS
^{0
is expanded it.
descending
powers of
?,
the coefficient of
^r1 i
s kr ; so that, as
m.
•no
it liolds for ■
^''j + h +
r = 0, 1, ...
, w
il.
we have
and therefore
/(?)=•(■«) If +1 +
.....?
+ .
...}.
As /(^) is a polynomial in f, all terms involving negative powers
of ^ must disappear, when multiplication is effected on the right
hand side ; and therefore
/(O ="S^ k, {^^'  (C.r"''^ + .. . + C^r,)],
the coefficient of k^i being unity. If therefore w and its first s
derivatives are aii to acquire the value zero when z=0, then the
degree of the polynomial f{^) is m — s — 2.
In order to obtain the customary expression for W, let the
contour be chosen so as to include all the zeros of ^(£f). Let a^
be a zero, and let its multiplicity be w,, so that
,(,(f).(!:.,)".f(r),
where the roots of ^i (^) are the other roots of ip (^). Let
/«)_ 41. +_^:_+ I ^y. ,/■«)
*«)"?".(?«■)■ (f".)*.(0'
jl'ii, ^'ai, ..., being constants, and /i (0 a polynomial of order
m — Ki — 1. So far as the first tii terms are concerned, their
contribution to the value of the expression for W is given by
taking a contour round a, only. We then have
2lih
Jf")
(rl)!' ^
on changing the constants; and therefore the part, arising through
the root Ki of multiplicity ni, in the expression for the integral is
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18
hermite's method for equations
[6.
involving a number of constants equal to the multiplicity of the
root. This forra holds for each root in turn ; and therefore the
number of constants is the sum of the multiplicities, that is, it is
equal to m, the degree of <l> (f ). But m is the number of arbi
trary constants in /(?), when it is initially chosen: these can
therefore be replaced by the constants A in the expression
S (jIj + ^=2 + . . . + A„£"") 6"^
the summation extendir
denoting the
occurs when a
another.
■ the roots a of (j){^) = Q, and n
lultiplicity of a. The simplest ease, of course,
the roots of ^{f) = are different from one
The method can be applied to the equation
where F(!) ia any function of s. Consider
where </)((;■ 1 a tl
in f with (u l,.n v.
integratioD esteu If
<l>(0 = 0. Then
/••'%?«
e as b fore / f) 15 a polvnumial
i coeftic ent^ f he powers ut f, and
■0 t u tl at icJei ill the louts ot
^™,r,/C,J)«Oi
li
n auceession, until we have
C^J{^,Odi=0;
(fo™
/{•,0'KO
/'■
■w*+/^)'""r/"'"«
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6.] WITH CONSTANT COEFFICIENTS 19
Heoce, remembering that /(s, is a polynomial in f and that therefore
we hiive W as a solution of the given equation if, in addition to the other
cocditioi:iK, which are that
forj=2, 3, ..., m, we have
Now as the contour embraj;es all the roots of <}> {0, we have*
for r = % .,., m ; ao that, taking
where d (s) ia a function of z at ouv disposa], we satisfy the ni — 1 formal
conditions unconnected with F(z) ; and then 6 (?) must be such that
But as
«»3i,f»
/(•,C)S'»+5i;./'«""^f(»)'i«,
and therefore
Hence
where j'(f)is, so far as concerns this mode of determining /(e, f), any function
of f, and integration with r^ard to u ia along any path that enda in s. When
F (e) ia zero, / (s, f ) reducea to ff (f) ; and then the solution of the differential
equation shews that ji(f)is a polynomial in £, of degree not higher than to — 1,
Aecordii^ly, as ff(^) is independent of i, we take it to be a polynomial of
degree n^— 1 in f, with arbitrary conatanta for the coefficients ; and then the
integral of the equation has the form
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20 CONTINUATION OF THE [6.
where the f t t t n est nda nd any mple contour including all the
roots of fji (i)=0 Tj d tl teg at n t nds from any arbitrary initial
point along J p tl (the pi th b ttcr) to 2.
Tho ai 1 nt gral th [ n f Wis clearly the complementary
function, and th d ul 1 mte^ri,! h p t ilar integral, in the primitive of
the differential equation. The expression can be developed into the customary
form, in the same way as in the simpler case when ii* (2) vanishes.
Hermite's investigation, based upon Cauchy's treatment by the calculus of
residues as espounded in the Exercices de Math^mMiqum, is given in a rnei
in Darboux's BvXl. des Sciences Math., 2"" S^r. t in (1879), pp. 311—325 :
followed by a brief note {L c, pp. 325 — 328), due to Dai'bous. A mci
by Collet, Ann. de I'tc Norm. Sup., 3™ Ser. t. IV (1887), pp. 129—144, may
also be consulted.
The Process of Continuation applied to the Synectic
Integral.
7. The synectic integral P(z— ^) is known at alt points in
the domain of J", being uniquely determined by the assigned
initial conditions at if. So long as the variable remains within
this domain, the integral at z does not depend upon the path of
passage from ^ to s, so that the path from f to z can be deformed
at will, provided it remains always within the domain. Let ^' be
any point in the domain ; then the values of the integral and its
first m — 1 derivatives at £" are uniquely determined by the initial
conditions at f, and they can themselves be taken as a new set of
initial conditions for a new origin ^'. Accordingly, construct the
domain of f ' ; and, with the values at f taken as a new set of
initial values, form the synectic integral which they determine.
As the new initial values are themselves dependent upon the
initial values at ^, the synectic integral in the domain of ^' may
be denoted by Pi (2 5^. 0
If the domain of ^' lies entirely within that of ^ (it then will
touch the boundary of the domain of ^ internally), the series
Pi (e — ^', must give the same value as P (z — ^): for every
point z in the domain of if' is then within the domain of f, and it
is known that the synectic integral is unique within the original
domain.
If part of the domain of if' lies without that of ^, then in the
remainder (which is common to tho two domains) the series Pi
must give the same value as P. But in that part which is
outside, the series Pi defines a synectic integral in a region where
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7.] SYNECTIC INTEGRAL 21
P does not exist ; it therefore extends our knowledge of the
integral, and it is a continuation of the synectic integral out of
the original domain.
Let Z be any point in the plane; and join Z ko i^ hy any
curve, drawn so as not to approach infinitesimally near any of
the singularities of the coefficients in the differential equation.
Beginning with if, construct the domains of a succession of points
along this curve, choosing the points so that each lies in the
domain of a preceding point and each new domain includes some
portion of the plane not included by any previous domain. Owing
to the way in which the curve is drawn, this choice is always
possible and, after the construction of a limited number of
domains, it will bring Z within a selected region. With each
domain we associate its own series ; so that there is a succession of
aeries, each contributing a continuation of its predecessor. We
can thus obtain at ^ a synectic integral of the equation, which is
uniquely determined by the initial values at f and by the path
from 5" to Z.
Further, taking the values of the integral and its first m — 1
derivatives at ^ as a set of new initial values, and taking the
preceding curve reversed as a path from Z to if, we obtain at if the
original set of assigned initial values. To establish this state
ment, it is sufficient to choose the succession of points along the
curve in the preceding construction, so that the centre of any
domain lies within the succeeding domain, and to pass back from
centre t<) centre. Stating the proposition briefly, we may say
that the reversal of any path restores the initial values.
By imagining all possible paths drawn from any initial point if
to all possible points z that are not singular, we can construct the
whole region of continuity of the integral, as defined by the
differential equation and by the initial values arbitrarily assigned
at if : moreover, we shall thus have deduced all possible values of
the integral at z, as determined by the initial values at if. It is
clear, from the construction of the domain of any point and after
the establishment of a synectic integral in that domain, which
can be continued outside the domain (unless the boundary of the
domain is a line of singularity, and this has been assumed not to
be the case), that the region of continuity of the integral is
bounded by the singularities of the coefficients. As has already
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22 DEFORMARLE [7.
been remarked, these singularities are called the singulanties of
the equation. Thus all the critical points of the integral are fixed
points ; and if the equation be taken in the form
l?™w
where the functions q^, ..., q^ are holomorphic over the finite part
of the plane and have no common factor, these critical points are
included among the roots of qo, with possibly z = 'Xi also as a
critical point. The value of the integral at an ordinary point near
a singularity has been obtained as a synectic function valid over
the domain of the point, which excludes the singularity. In
later investigations, other expressions for the integral at the
point will be determined, when the point belongs to a different
domain that includes the singularity.
8. Any path from ^ %o z can be deformed in an unlimited
number of ways : and it is not inconceivable that these deforma
tions should lead to an unlimited number of values of the integral
at z, as determined by a given set of initial values : but the
number is not completely unlimited, because all paths from ^toz
lead to the same final value at z with a given set of initial values at
£", provided they are deformable into one another without crossing
any of the singularities. To prove this, consider a path from f to
z, drawn so that no point of it is within an infinitesimal distance
of a singularity, and draw a second path between the same two
points obtained by an infinitesimal deformation of the first; no
point of the second path can therefore be within an infinitesimal
distance of a singularity. On the first path, take a succession of
points zi, z^, ..., so that 3i lies within the domains of % and of z^,
% within the domains of z^ and ir,, and so on. On the second path,
take a similar succession of points a/, si, ..., near Si, i^a, ... respec
tively, in such a way that s/ lies in the part common to the
domains of ^ and s,, while z, is in the domain of zl\ Sj' in the
part common to the domains of z, and z^, while z^ is in the domain
oi si; and so on. Join ziZl, z^^, ... by short arcs in the form of
straight lines.
Now we have seen that, in any domain, the path from the
centre to a point can be deformed without affecting the value of
the integral at the point, provided every deformed path lies within
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8.] PATHS 23
the domain. Hence in the domain of ^, the path f^i gives at ^,
the same integral as the path ^z^'^i. This integral furnishes a set
of initial values for the domain of Sj ; and then the path S]Zs gives
at % the same integral as the path siS^s^z^. Consequently the
path fsi^2 gives at z^ the same integral as the path fs/^j, followed
by ztZjZ^Zi. But the effect of z^Zi followed at once by z^z^' is nul,
because a reveraed path restores the values at the beginning of
the path ; and therefore the path i^z^z^ gives at z^ the same integral
as the path t^z^ziz^. And so on, from portion to portion; the last
point on the first path is z, which also is the last point on the
second path; and tlierefore the path tz,z^...z gives at z the same
integral as the path t^(z^...z.
Now take any two paths between if and z, such that the closed
contour formed by them encloses no singularity of the equation.
Either of them can be changed into the other by a succession of
infinitesimal deformations : each intermediate path gives at z the
same integral as its immediate predecessor: and therefore the
initial path and the final path from ^ to s give the same integral
at z ; which is the required result.
If however two paths between ^ and z are such that the closed
contour formed by them encloses a singularity of the equation,
then at some stage in the intermediate deformation the curve will
pass through the singularity, and we cannot infer the continuation
along the curve or the deformation into a consecutive curve as
above. It may or may not be the case that the two paths from
X,i>a z give at z one and the same integral determined by a given
set of initial values ; but we cannot assert that it is the case.
Accordingly, we may deform a given path without i
the integral at the final point, provided no singularity is c
in the process. Moreover, in order to take account of different
paths not so deformable into one another, it will be necessary to
consider the relation of the singularities to the function represent
ing the integral : this will be effected in a later investigation.
When two paths can be deformed into one another, without
crossing any singularity, they are called reconcileable ; when they
cannot .so be deformed, they are called irreooncileable. If two
irreconcileable paths lead at z to different integrals from the same
initial values at f, the closed circuit made up of the two paths
leads at ^ to a set of values different from the initial values.
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24 FORM OF THE [8.
These new values can be taken as a new set of initial values :
when the same circuit is described, they are not restored, so that
either the old initial values or a further set of values will be
obtained : and so on, for repeated descriptions of the ciicuit. By
this process, we may obtain any number, perhaps even an unlimited
number, of sets of values at ^ deduced from a given initial set ;
and thus there may be any number, perhaps even an unlimited
number, of values of the integral at any point e.
Consider any path from f to s ; and without crossing any of
the singularities, let it be deformed into loops, drawn from ^ to the
singularities and back, (these loops coming in appropriate success
ion), followed by a simple path (say a straight line) from f to s.
The final value of the integral at z is determined by the values
at f at the begiuning of the straight line, and these values are
deducible from the initial values originally assigned. Hence the
generality of the integral at z is not affected by taking any particular
path from f to z, provided complete generality he reserved for the
initial values : and therefore, from this aspect, it will be sufiBcient
to discuss the complete system of integrals as arising from com
pletely arbitrary systems of initial values at an ordinaiy point.
This investigation relates to properi^ies of the integrals, which will
be found useful in discussing the effect of a singularity upon a
given integral ; it will accordingly be underiiaken at once.
9. It has already been remarked that the synectic integral,
determined by the arbitrary constants which are assigned as the
initial values of the function and its derivatives, is linear and
homogeneous in those constants: so that, if /a,,, jUia, .., fiim denote
the arbitrary constants, and w^ denotes the synectic integral which
they determine in the domain of au ordinary point i^, we have
where ii,, Mj, .... m™ are holomorphic functions of 3 — t^, not involv
ing any of the arbitrary coefiicients /i. Take other m — 1 sets of
arbitrary constants fi, such that the determinant
I ".I , ^i. Mm , =A(Osay,
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9.] SYNECTIC INTEGRAL 25
is different from zero. Each set of m constants, regarded as a set
of initial values, determines a synectic integral, in the domain of
^; as the quantities v^, u^, ..., u^ in the expression for Wi do not
involve the arbitrary constants determining w^, it is clear that the
expressions for these other m — 1 integrals are
Let Msi denote the minor of jist in the nonvanishing determinant
A(?); then from the expressions for the m integrals w,, ..., w^ ^"^
terms of Wi, ..., u^, we have
^.{^)u=^MuVh + M^t'a},^+M^ty'm, (t=l,...,m).
Now any other synectic integral, determined in the domain of ^
by assigned initial values $i, B^, ..., 5,„, is given by
where the constants & are given by
:«,!
(r1,
».).
These constants S cannot all vanish, when the constants ^i, 9^, ...,
6m are not simultaneous zeros : for the determinant of the minors
Mri is {A (?)™~', and therefore is not zero. Accordingly, any
integral can be expressed as a linear combination of any m
integrals, provided the determinant of the initial values of those
m integrals and their first m — 1 derivatives does not vanish. But
it is not yet clear that the integrals w^, ..., w,„ are linearly inde
pendent of one another; until this property is established, we
cannot affirm that the expression obtained is the simplest obtain
able.
Consider therefore, more generally, the determinant of the m
integrals and their first m~l derivatives, not solely at f but for
any value of s in the domain of ^, say
A{z)^.
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26 A
PECIAL
DETERMINANT
[9.
When 2= f, it becomes the determinant of initial values denoted
by a (0 We have
d^^^)
d'"iu,
d™''^'Wi
dt 
d^ •
dz'^i *"'
d^Wi
rf^^i/l.
d!f •
d^" "'■
d"*wm
i— %«
T^
d»» "'
=P.aW
on substituting tor ^,...,
~^™ their values in terms of the
derivatives of lower orders as given by the equation. Hence
Now within the domain of i^, the function jj, is regular, being of
the form P,{z — ^); hence the integral in the exponent of e is of
the form R{z — t,), where it is a regular function that vanishes
when z=^t. Consequently the exponential term on the righthand
side does not vanish at any point in the domain of 5'; also A(f)
is not zero ; so that A (z) has no zero within the domain of f.
Moreover, each of the quantities w^, ..., w„ is a holomorphic
function of z — ^ in that domain, so that A{z) is holomorphic
also; hence A(3) has no zero and no infinity within the domain
of the ordinary point C
i (z) may vanish
a any region of
Ak a matter of fact, the only points where
become infinite are the singularities of pj. For
3 of the functions w,, ..., !(i,„, we have
4j,)_ jji*
4(0
the path from f to s lying within that region, v/hile s is not i
the domain off. If a be one of the singularities of pi, the expression of pj i
any part of an annular region round a as centre is of the form
where the number of terms in
according as the singularity ii
f sa is finite or infinite,
jntial ; and g (s) is hoio
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9.] FUNDAMENTAL SYSTEMS
morphio in the vicinity of a. Taking the simplest c
ix^=a^= ... =0; then
.'=•*(!)>
shewing that a ia a zero of A (2) if the real part of a, be positive, and that it
is an infinity of A iz) if the real part of a, be negative. More generally, the
nature of A (2) in the vicinity of any singularity a depends upon the character
of Pj in that vicinity : in tte case of the above more general form, a is an
essential singularity of A (2).
Fundamental Systems of Integrals.
10. The linear independence of w,, ..., Wm, and the property
that A {£) has a finite nonzero value at any point in the plane
■which is not a singularity of the equation, are involved each in
the other.
It is easily seen that, if a homogeneous linear relation between
Wi, ..., win, of the form
CiWi + . . , + c^'.y™ =
were to exist, the quantities c,, ..., c^ being constants, then A {z)
would vanish lor aii values of s. The inference is at once
established by forming the m — 1 derived equations
and eliminating the m constants Ci,..., c,„ between the m equa
tions which involve them linearly: the result of the elimination is
A(.)0.
Hence if, for any set of integrals Wj,..., w™, the determinant A{3)
does not vanish (except possibly at the singularities of the
equation), no homogeneous linear relation between the integrals
exists.
To establish the inference that, if A (a) does vanish for all
values of z, a homogeneous linear relation between Wi, ..., w^
exists, we proceed as follows.
In the first place, suppose that some minor of a constituent in
the first column of A {«), e.g. the minor of —j J^ in A {z), say
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28 A FUNDAMENTAL SYSTEM [10,
Ai ie), does not vanish for all ordinary values of 2 ; and take m
quantities j/i, ..., ym, the ratios of which are defined by the
relations
From the hypotheses that A (s) = and that Ai does not vanish, it
follows that
' ds""'
Because of the assumption that A^ does not vanish, the ratios
2/™' ym' '"' 2/m
are determinate finite functions of s.
Differentiate the first of the relations: then, using the second,
we have
j>i + ...+)/Jw« = 0,
where 3// denotes dyffdz, for the n values of r. Differentiating
the second of the relations, and using the third, we have
, dw, , dw,„
and so on, up to
obtained by differentiating the last of the postulated relations
and by using the deduced relation. We thus have m — 1 relations,
homogeneous and linear in the quantities y^, ., y^ \ in form,
they are precisely the same as the m — 1 relations, which are
homogeneous and linear in the quantities j/i, ..., i/,„. Hence, as
A, does not vanish, we have
*=.&',, (,.= 1,2 ml).
;©=
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10.] AND ITS DETERMINANT 29
SO that
— ^constant = ,—''■, (j = 1, 2, .... m — 1),
where Xi, ..., \a~i> ^m ^^^ simultaneous values of y,, ..., ym\> ym
for any particular value of s : that is, the quantities X are con
stants. This particular value of s is at our disposal; we may
assume that X™ is different from zero, because the ratios of ^i , . . . ,
i/ni_, to ym are determinate and finite. Now
hence
XlW, + . . . 4 X^Wm — 0,
that is, a linear relation exists among the quantities w, if A (z) is
zero, and some minor of a constituent in the first column does not
vanish.
Next, suppose that the minor of every constituent in the first
column vanishes : in particular, let Ai {z) = 0, for all ordinary
values of z. Then A, (z) is a determinant of m — 1 rows and
columns, constructed from m — 1 quantities Wi, ..., w„^^ in the
same way as ^{z), a determinant of m rows and columns, is
constructed from the m quantities w,, ..., Wm The preceding
analysis shews that, if some minor of a constituent in the first
column of Ai (z) does not vanish for all ordinary values of z, then
a relation
where k, ,...,«nii ^^^ constants, is satisfied: so that a linear
relation exists among the quantities w, and it happens not to
involve ic^
Lot the process of passing from A {z) to A, (s), fiom A, {z) to a
corresponding minor, and so on, be continued; the successive
steps are effected by removing the successive columns in A{3)
beginning from the left and by removing a corresponding number
of rows. At some stage, we must reach some minor which is not
zero tor all ordinary values of e : so that
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30 THE NUMBER OF [10.
vanishes whea s = 0, 1, ..., r, but is different from zero when
s = r + l. Then the earlier analysis shews that a linear relation
of the form
/JlW, + . , . + pm^VJr„^, =
exists, where pj,..,,p„^, are constants: in effect, a linear homo
geneous relation among the quantities Wi, ..., Wm which happens
not to involve Wmi+i, ■■, w™ Hence, if the determinant A(^),
constructed from ike m integrals w^, ..., w^, vanishes fiyr all
<yrdinary values of z, there is a komoffeneo'us linear relation between
these integrals.
Integrals are sometimes called independent when they are
linearly independent, that is, connected by no homogeneous linear
relation ; but the independence is not functional, because all the
integials are functions of the one variable z. A set of m linearly
independent integrals w is called a fundamental system. ; and each
integral of the set is called an element or a member of the system.
The determinant A (a), constructed out of a set of m integrals, is
called the determinant of the system; so that the preceding results
may be stated in the form ; —
If the determinant of a set ofni integrals vanishes for ordinary
(that is, nonsingular) values of the variable, the set cannot constitute
a fundamental system ; and the determinant of a fundamental
system does not vanish for any nonsingular value of the variable.
11. We now have the important proposition: —
Every integral, which is determined by assigned initial values,
can be expressed as a homogeneou^s linear combination of the
elements of a fundamental system.
Let W denote the integral determined by the assigned values
at if, taken to be an ordinary point of al! the coefficients in the
differential equation ; and let w, , . . . , Wm he a fundamental system.
Let constants c,,...,Cm be deduced such that, when 3 = ^, we
have
W= 2 CaWa \
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11.] LINEARLY INDEPENDENT INTEGRALS 31
This deduction is uniquely possible; because the determinant of
the quantities c on the righthand aides is the determinant of a
fundamental system, and therefore does not vanish when z=^.
Thus W ~ i, Ct,WK is an integral of the equation; this integral
and its first m — 1 derivatives vanish when 3=^; so that it
vanishes everywhere (Cor, I, § 5), and therefore
the constants c being properly determined as above.
Coe. I. Between any m + 1 branches of the general solution,
there must be a homogeneous linear relation. For if m of them be
linearly independent, the remaining branch can be regarded as
another integral : by the proposition, it is expressible linearly in
terms of the other m.
Cor. II. Any system of integrals u^, ..., «,„ is fundamental if
no relation exists of the form
where A^, ..., A^ are constants. For taking a fundamental system
W], ..., Wm, we can express each of the solutions u in the form
Mr = «lr«'l+ ■■■ + IhnrVm. (r=l, 2, .,., m),
where the coefficients a are constants. If G denote the determ
inant of these m^ coefficients, C must be different from zero: for
otherwise, on solving the m equations to express Wi in terms of
Ml, ,.., Mfli, we should have a relation of the form
A^Uj+ ... + .4„M„ = Cwi = ;
and no such relation can exist. If, then, A^ (e) denote the deter
minant of the set of integrals u, and if A„ (s) denote that of the
fundamental system w^, ..., Wm, we have
A.WCA,W,
by the properties of determinants. Now C does not vanish, nor
does A,j (s) at any ordinary point in the plane ; hence A„ (2) does
not vanish at any ordinary point in the plane, and therefore
Ui, ...,u,a ^^e a fundamental system of integrals.
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32 A SPECIAL [11.
The result may be stated also as follows : If m integrals u be
given hy equations
u =a,iL,+ +« ' „ <<=1 m)
where the deteimmant of the coe_ffnienfs a i/, nd ^eio and t/e
integrals w are a fjnda mental system then the bystem of mteQiaU
u is also fundamental
12. One paiticuHi fundamental sjstem for the difteiential
equation can be jbti.ined as follows Let w be a sppciil mtegial
of the equation that is an integral deteimioed 'h^^ an> special set
of initial conditions, and substitute
w = wjvde
in the e(juation : then v is determined by the equation
Similarly, let v^ be a special integral of this new equation, with
the appropriate conditions ; then substituting
V = vjudz,
we find that the equation, which determines u, is of the form
where
mldv,
'■^ = ^^ ^rf7
And so on.
It is manifest that the quantities
w,, wjv^dz, w,f(vjihds)ds, ...
are integrals of the original equation. Moreover, they constitute
a fundamental system ; for, otherwise, they would be linearly
connected hy a relation of the form
CiW, + dwjvjds + C3W,j{vJu,dz) dz+ ... = 0,
that is, ,
Cj + c.jjv,dz + cj(vju,dz) dz+ ... = 0.
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12.] FUNDAMENTAL SYSTEM 33
When this is differentiated, it gives
CjUi + Civjujds + ... = 0,
that is,
c.i + c,Juidz + ... = 0.
Effecting m — 1 repetitions of tliis operation of differentiating and
removing a nonzero factor, we find
as the result at the last stage. Using this in connection with the
equation at the last stage but one, we have
c™, = 0.
And so on, from the equations at the various stages, we find that
all the coefficients c vanish. The homogeneous linear relation
therefore does not exist : the system of integrals, obtained in the
preceding manner, is a fundamental system.
As an immediate corollary from the analysis, we infer that
^i. Vifu,dz,...
constitute a fundamental system for the equation in v ; and so for
each of the equations in succession.
The determinant of this particular fundamental system is
simple in expression. Denoting it by A, and denoting by Ai the
determinant of the fundamental system of the equation in v,
we have, as in § y,
^Pi'
IdA
i^'dz
1 dAi _ _ m dwi
A dz Ai dz '
where Xi is a constant. Similarly, if A3 denote the determinant
of the fundamental system of the equation in u, we have
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34 FORM OF THE DETERMINANT [12.
and so on. The last determinant of all is the actual integral of
the last of the equations ; hence
iCwi™?)j"''u,™^..,
where (? is a constant. Moreover, A is the determinant of a par
ticular system, so that C is a determinate constant. It is not
difficult to prove that
and therefore
conaequtntly,
Sx. Verify the last result, as to the form of A, in the case of
(i) Legendre's equation :
(ii) the equation of tte hjpei^eometric series :
(iii) Bessel's equation.
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CHAPTER II.
General Form and Properties of Integrals near a
Singularity.
13. We have seen that, within the domain of an ordinary
point, a synectio integral of a linear differential equation is
uniquely determined by a set of assigned initial values ; and that
the said integral can be continued beyond that domain, remaining
unique for all paths between the initial and the final values of the
variahle which are reconciteable with one another. When the
variable is permitted to pass out of its initial domain though
returning to it for a iinal value, or when two paths between the
initial and the final values are not reconcile able, the various
propositions that have been established are not necessarily valid
under the modified hypothesis : it is therefore desirable to con
sider the influence of irreconcileable paths upon an integral,
still more upon a set of fundamental integrals. Remembering
that any path is deformable without affecting the integral if, in
the deformation, it does not pass over a singularity, we shall
manifestly obtain the effect of a singularity, that renders two
paths irreconcileable, by making the variable describe a simple
circuit, which passes from the point z round the singularity
and returns to that point z, and which encloses no other
singularity.
Let a be the singularity round which the simple closed circuit
is completely described by the variable. Let m;,, ..., in^ denote a
fundamental system at z ; and suppose that the effect of the
3—2
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36 EFFECT OF A SINGULARITY [13.
circuit is to change the m integrals into w/, ..., Wm respectively.
That the set of m new integrals thus obtained is a fuudamental
system can be seen as follows. If it were not a fundamental
system, some relation of the form
2 k,w; =
would exist, with constant coefficients k, for all values of z in the
immediate vicinity. In that case, the quantity S krW,' (which is
an integral) is zero everywhere, together with all its derivatives,
as it is continued with the variable moving in the ordinary part
of the plane. Accordingly, let the integral be continued from z
along the closed circuit reversed until it returns to z where, by
what has been stated, it is zero. The effect of the reversal is
(§ 7) to change w/ into w, : and so the integral after the reversed
circuit has been described is % krWr, so that wc should have
2 Kw, = 0,
contrary to the fact that v\, .,,, w^ constitute a fundamental
system. The initial hypothesis from which this result is deduced
is therefore untenable : there is no homogeneous linear relation
among the quantities w/, ..., Wm, which therefore form a funda
mental system.
Since the system Wi, ..., Wm is fundamental, each of the inte
grals w/, ..., Wm is expressible linearly in terms of the elements
of that system ; so that we have equations of the form
w/ = a:siiy,+ ...l«^w™, (s=l, ...,m),
where the coefficients a. are constants. As the system Wg is
fundamental, the determinant of these coefficients is different
from zero : this being necessary in order to ensure the property
that W], ..., Wm are expressible linearly in terms of w/, ..., Wm, a
fundamental system.
Take any arbitrary linear combination of the system, say
where the coefficients p are disposable constants ; and denote this
integral by u. When the variable desci'ibes the complete closed
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13.1
UPON A FUNDAMENTAL SYSTEM
3Y
circuit round the singularity, let w' denote the modified value of
u, so that
U = pjWi + . . . + pmlOm'
= Pi 2 a,rW,
. + pml a^rWr
It is conceivable that the coefficients p could be chosen so that
the integral reproduces itself except as to a possible constant
factor; a relation
would then be satisfied, 6 being a constant quantity. This rela
tion, in terms of w,, ..., w™, is
Pi 2 a^rWr^ ... + pm 2 a^rWr
= 9 (piW, + ... + p™Wm),
which, as it involves only the members of a fundamental system
linearly, must be an identity: the coefficients of w,, ..., w™ must
therefore be equal on the two sides. Hence we have
Pittis + p2 (0=2  ^) 4
■ + PmCtmi
■ ■ + pm«mi
If, therefore, $ be determined as a root of the equation
the preceding relations then lead to values for the ratios of the
constants p for each such root. It is to be noted that, in this
equation, the term, which is independent of $, does not vanish,
for it is the determinant of the coefficients a ; hence the equation
has no zero root.
As the equation definitely possesses roots 9, it follows that
integrals exist which, after a description of the simple contour
round a, reproduce themselves save as to a constant factor. If it
should happen that the constant factor is unity, then the effect of
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38 THE FUNDAMENTAL EQUATION [13.
description of the contour upon the integral is merely to leave it
unchaiiged : in other words, such an integral is uniform in the
vicinity of the singularity.
Propeuties of the Fundamental Equation.
14. The special significance of the equation, in relation to
the singularity a, lies in the proposition that the coefficients of ike
various powers of 6 in A =0 are independent of the fundamental
system initially chosen for discussion. To prove* the statement,
it will be sufficient to shew that the same equation is obtained
when another fundamental system is initially chosen. For this
purpose, let y,, ,,.,ym denote some other fundamental system;
and suppose that, by the simple closed contour round a described
by the variable, the members of the system become j^/, ,.,, ym
respectively. Then, as both these systems are fundamental, there
are relations of the form
y/ = 0s,yi +  + 0^y^, (s = i, ....m),
where the determinant of the coefficients /3 is not zero. The
equation B = 0, corresponding to j1 = for the determination of
the factor 8, is formed from the coefKcients ^ in the same way
as A from the coefficients a, so that the expression for B is
B =
Because each of the sets w„ ,.., w^; i/,, ,.,, i/^; is a funda
mental system, the members are connected by relations of the
form
y., = y„w, + y^w^+... 17™^^, (r^l, ..., m),
where the determinant of the coefficients, which may be denoted
by r, is different from zero. The quantity
is zero everywhere in the vicinity of z ; and it is an integral,
which accordingly is zero everywhere in its continuations over the
* The proof adopted is due to Hamburger. Crelk, t.'Lxxvi (1873), pp. 113—125.
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14.] IS INVARIANTIVE 39
ordinary part of the plane. When it is continued along the
simple contour round a, the variable returning to z, the integral
is zero there ; that is,
Hence
and therefore
S 2 /SrsJeiWi =22 7rs«siW[.
This relation involves only the members of a fundamental system
hnearly ; hence it must be an identity. We therefore have
2 ^r67B(= 2 7rsas*
say, the relation among the constants holding for all values of r
and t. Now forming the product of the determinants P and A,
we have
7n. Tis' 7i3.
721. 722. 7S3'
7si, 7aa, 7as.
5ii7u^. Sk7iA ...
821 — 7h^, S22 — 7,2^, ■ ■ ■
say ; and similarly, forming the product of B and V, we have
' — 6, ^,2 , /3i3 , .■■ 7ii, 7si, 731.
, ji^0, /3a , ... 7i3. 7^. 732.
, ^^ ,0x1 — 0. ■■■ 7i3> 723. 73s.
7n^. S,.,7,A ...
72A ^^y^0, ...
= 0,
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40 INVAEIANTIVE PROPERTIES [li.
identically. Also F does not vanish ; hence
for all values of B.
Accordingly, the equation A = is invariantive for all funda
mentaJ systems in regard to the effect of the singularity a upon
the memhers of the system: it is called* the fundamental equation
belonging to the singularity a. We note that its degree is equal to
the order of the differentia! equation.
While the equation is thus invariantive for all fundamental
systems, the actual invarianee of one of its coefficients is put in
evidence, either when the differential equation of § 2 is initially
devoid of the term involving j;^ , or after the equation has
been transformed by the relation
SO as to be devoid of the term involving , ^_^ . In A =0,the
term, which is independent of 6 is equal to unity, a property first
noted by Poincar^f. For when ^i is zero, the determinant A of
the fundamental system is a constant, for (§ 9) its derivative
vanishes ; it therefore is unchanged when the variable describes
a simple closed circuit round the singularity. The effect of such
a circuit upon A is to multiply it by the term in A which is
independent of 8 : accordingly, that term is unity.
The linear equation can always be modified so that the term
involving the derivative of the dependent variable next to the
highest is absent; and the necessary linear modification of the
dependent variable leaves the independent variable unaltered.
This change does not influence the law giving the effect, upon
the integrals, of a description of a loop round the singularity;
and the fundamental equation is independent of the choice of
the fundamental system. Accordingly, the coefficients of the
various powers of 9 (except the highest, which has a coefficient
(—1)™, and the lowest, which has a coefficient unity) are fre
quently called the invariants of the singularity : they are m — 1
in number.
• Sometimes also tha charaeteristic equation.
t Acta Math., t. iv (1884), p. 202.
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15.] OF THE FUNDAMENTAL EQUATION 41
15. There is a further important invariantive property of the
determinants A(d), B{ff), viz.: If all minors of order n (and
therefore all minors of lower order) in A (8) vanish for a particular
value of 6, but not all those of order n + 1, then all minors of order
n in, B {&) also vanish for thai, value of 8, hut not all those of order
n + \.
A minor of order n is obtained by sappresaing n rows and n
columns ; accordingly, the number of them is
say. Let them be denoted by ftfj, hij, c^, dij when formed from
A {0), B (0), r, J) respectively, where * and j have the values
1, ..., /i, these numbers corresponding to the various suppressions
of the rows and the columns. Then, regarding D as the product
of A and V, we have*
dij = Ci,aj, + Gjaaji + . . , + Ci^aj> ;
and regarding D as the product of B and T, we have
dij = biiCj, + bi^Cji+ ... +i'i„Cj„.
All the quantities (ifj are supposed to vanish for a particular value
of d ; hence for that value all the quantities dij vanish. Assigning
to j all the values 1, ..., ^ in turn, we therefore have
= Cii&ii +Cis&i3 + ... +c,^bi^ \
= cAi + Cs^bi2 + ... +c.^bi^ I ,
= C^ibij + C^s6;s + . . . + c^^bi^ I
The determinant of the coefficients of bi,, hi,, ... , V '^ equal tof
T\
where
x = _C^i)'_.
{mn\)\ nV
that is, the determinant does not vanish. Accordingly, we must
have
ht^^O, b^ = (},...,hi^ = 0;
as this holds for all values of i, it follows that all the minors ot
B{9) of order n vanish for the particular value of 0.
* Scott's Determinants, p. 53. + ih., p. lil.
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42 ELEMENl'ARY [15.
The minors of B (S), which are of order li + 1, cannot all vanish
for the value of 0; for then, by applying the result just obtained,
all those of A (0), which are of order n+1, would vanish, contrary
to hypothesis.
16. A more general inference can bo made. Leaving arbi
trary and not restricting it to be a root of the fundamental
equation, the two expressions for dij give
holding for all values of i andj. Taking this equation for any one
value of _y and for all the /t values of i, we have ^ equations in all,
expressing a,,, a^, ..., iij> linearly in terms of bpq. The determi
nant of coefficients on the lefthand side is T*, as before, and does
not vanish ; so that each of the quantities Oj^ is expressible linearly
in terms of the quantities ftp,, the coefficients involving only the
constituents of V. Similarly, taking the equation ibr any one
value of i and for all the ^ values of j, we find that each of the
quantities bp^ is expressible linearly in terms of the quantities aj>,
the coefficients involving only the constituents of F. If therefore
all the quantities a^r have a common factor — 0i, and if that factor
be of multiplicity o, then all the quantities bpq also have that
factor common and of the same multiplicity a ; and conversely.
These results associate themselves at once with Weierstrass's
theory of elementary divisors*. If {0 — Oi)" is the highest power
of 8 — 0, in A (6), if (6 — O^y^ is the highest power of that quantity
common to all its minors of the first order, if (0 — 01)"' is the
highest power common to all its minors of the second order, and
so on, then (as will be proved immediately)
a >a;><7^>...;
and
(«»,)—', (9 e,).., ...
are called elementary divisors of the determinant A (0). It follows
from the preceding investigation that the elementary divisors of
the fundamental equation, are invariantive, as well as the equation
* Berl. Monatsber., (1868), pp. 310—388; Oes. Werke, t. ir, pp. 19—44. See
aUo & memoir by Sauvage, Ann. lie l'£c. Norm., 3" S&., t; vm (1891), pp. 26a— SiO ;
and a treatise by Math, EtemenlaTtlieiUr, (Leipzig, 1899).
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16.] DIVISORS 43
itself; for they are independent of the particular choice of a
fundamental system.
If the earliest set of minors of the same order that do not all
vanish when = Si is of order p, so that they are of degree m.~ p
in the coefficients in A, then the elementary divisors are
being p in number : and then p is one of the invariantive numbers
associated with the particular singularity of the equation.
As two of the properties of the invariantive equation, associated with the
elementary divisorB, are required, they will he proved here : for full discusaion
of other properties, reference may bo made to the authorities quoted.
It is easy to obtain the result
cr>oi>irj> ..,,
just stated above. For
9^ Z A
M = .!/
wlieie J r 1^ the muioi ot a^, 6 In A„ there is a factor (00,)"', for each
of tlie qmiititiei ■!„ is i fir«t minor ; therefore that factor occurs in their
sum and, owm^ to the uimbination of terms, it may have an even higher
indes than o, On the left, the factor in ^  ^j has the index o  1 i hence
that is,
Similarly for the other inequalities.
Again, we know • that any minor of degree p which, can be formed out of
the first minors of A {&) is equal to the product of ^p^' {6) by the comple
mentary of the corresponding minor of A {6). Hence, taking p = 2, we have
relations of the form
A^B2A^B^ = AC,
ora of the first order, and C is a minor of the
r of the second order which is divisible by no
higher power of 6— 6, than (fl— fl,)"'; the lefthand side is certainly divisible
by {6  S^f"', and it may be divisible by a higher power if the terms combine ;
hence
that is,
Similarly, we have the other inequalities of the set
cro^>T^ir^5:a2irs>...><ri,_i,
so that the indices of the elementary divisors, as arranged above, form a
series of decreasing numbers.
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BRANCHES OF AN ALGEBRAIC FUNCTION"
[17.
Association of Differentiai Equations with Algebraic
Functions.
17. Before considering the roots of the fundamental equation,
it is worth while establishing a converse* of the propositions in
§ 13, as follows:
Let j/i, ,.., ym he m linearly independent functions of s, which
are uniform over any simplyconnected area not including any
critical point of the functions : let the critical points be isolated and
let each of them be such that, when a simple contour enclosing it is
described, the values of the Junctions at the completion of the contour
are given by relations of the form
yr'=a«,?i + . + «™2/m, ('• = 1, ...,m),
where the determinant of the coefficients a. is not zero, and the
constants may cJiange from one critical point to another : then, the
m functions are a fundamental system of integrals for a linear
differential equation of order m unth uniform coefficients.
It is clear that, if the functions are integraSs of such an equation,
they form a fundamental system because they are linearly indepen
dent. On account of this linear independence^ the determinant
dz'"^' '
d^"^"
d^^' ' dz"'' ' ■■■■
does not vanish for all values of z. Let As denote the determinant
which is derived fiom A by changing the stii column into j—^ ,
'"' rir^ ' ^^^ consider the quantity
For any contour that encloses no critical point, A and A, are
uniform, so that ps is uniform for such a contour. For a simple
contour, which encloses the critical point a and no other, the
' It is given by Tannery, Ann. de VKc. Norm.. Ser. 2"", t. iv (1875), p. 130.
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17.] AS A FUNDAMENTAL SYSTEM 45
determinant A after a single description acquires a constant factor
R, where R is the (nonzero) determinant of the coefficients in the
set of relations
^/r' = «rii/i+ + a™3/m, (r = 1, ...,m).
The determinant Aj acquires the same factor R, in the same
circumstances ; and therefore ps is unchanged in value by a
description of the contour, that ia, it is uniform for such a
contour. As this holds for each contour, it follows that pg is
uniform over the plane.
The m quantities y,, ..., y^ evidently are special integrals of
the equation
which is linear and the coefficients in which have been proved
uniform functions of s.
Corollary. If all the critical points of the functions are of
an algebraic character, that is, of the same nature as the critical
points of a function defined by an algebraic equation, and are
limited in number, then the uniform coefficients p m the differential
equation are rational functions of z. For as p^ is uniform, the
critical point a is either an infinity, or an ordinary value (including
zero). If it is an infinity, it can be only of finite multiplicity;
for the critical point is one, where A and A, can vanish only to
finite order because of the hypothesis as to the nature of the
critical point: that is, the point is then a pole of finite order.
Likewise, if it is a zero, the multiplicity of the zero is finite.
This holds at each of the critical points of the functions y\,..,ym\
and the number of such points is finite. Moreover, every point
that is ordinary for each of the functions is ordinary for A and Ag
and, in particular, A cannot vanish there : so that no such point
can be a pole of any of the coefficients p. It therefore follows*
that each of these coefficients is a rational meromorphic function
oiz.
The converse of the corollary is not necessarily (nor even
generally) true : it raises the question as to the tests sufficient
and necessary to secure that the integrals of a linear equation with
rational coefScients should be algebraic functions of the variable.
This discussion must be deferred.
* T. F., S 43.
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ALGEBRAIC FUNCTIONS
[ir.
46
Ex. 1. The most conspicuous instance arises when the dependent vari
able w is an algebraic function of z, defined bj an algebraic ec[iiation
of degi'ee m io w. Ba«h branch of the function so defined is uniform in the
vicinity of an ordinary point ; in the vicinity of a branchpoint, the branches
divide themselves into groups ; and any linear combination of them is subject
to the foregoing laws of change (which take a particularly simple form in this
ca.se) when z describes a circuit round a branchpoint.
To obtain the homogeneous linear equation of order m which is satisfied
by every root of j''=0, we can proceed as follows. Let ^(s) = be the
eliminant of /=0 and =~=0; so that* all the branchpoints of the alge
braic function are included among the roots of 0=0, though not every root is
a branchpoint. By a result f in the theory of elimination, we know that the
resultant of two quantics u and v of degi'ee ni. and n respectively in a variable
to be eliminated is of the form
where Mj and % are of degrees m 1, «— I respectively in that variable ; and
therefore
where i/ is of degree m— 2 io w and V is of degree ni I in n; But / is
? equal to zero for all the values of w considered ; hence
.ally derivable from Sjl
of the ehminant, (
To the last column, add the first column multiplied by .r™"'^"^, the second
multiplied by iii'"+"2, and so on : a change which does not affect the value of E.
The couBtituants in the new last column ace
x'^hi, x'^ht, .... xu, K, x^^^i), i™2i), .,., XV, v;
eipanding E by taking every term in this last column with its minor, eoUeo ting all
the terms involving ti into one set and those involving v into naother, we have
where v,
s of degree m 1 ii
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17.]
AND DIFFERENTIAL EQUATIONS
By m.»
i of/=0 which ia of degree m
in w, we can
, reduce v'f
contains i
10 power of w h^her than the (i
mI)th, say
where P, is a polynomial in w of degree not higher than m1. (If the
highest term in / has unity for its coefficient, then P, ia a polynomial in s
also.) Aj;ain,
d^ Pi dp, 1 dP, F, a^
d^a 02(3) ^ +0(2) g^ 02(5) a^
on reduoing to a common denominator ; by u
Pj can he made of degree not higher than nj
uniform functions of z. And so on, up to
where I'^ is a polynomial in i
being uniform functions of e.
ns of /=0, the polynomial
1 M, and its coefficients aro
7 of degree not higher than m  1, the coofflcionta
We thus have
Among these m equations t
m 1 quantities u^", a)*, ..., iP"
the form
can, by a linear combination, eliminate the
■^ irom the lefthand aides ; and the result has
&•«='
™^2m 9"
where Q„, <2i, ..., §m are uniform functions of s, Thia is satisfied for every
root JO of the algebraic equation : and it is of order m.
Corotlary. There ia one special case, when the differential equation is of
order m  1, viK., when the algebraic equation ia
/=w"'la2)«™=+ ... +0^=0,
BO that the term in w™~' is absent. We then have
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48 EXAMPLES OF [17.
SO that one of tho in, branches w can he espreaaed ]inearly in terms of the
others ; Tannery's r^ult ahewa that the differential equation is theii, of order
•not higher than m — 1. In that case, it would be sufficient to take only the
m — 1 equations
^1"*'. ( •»')•
For instance, consider the algebraic equation
u^ + to^M,
where u is any function of 2 ; it is to be expected that the liaear differential
equation satisfied by each of the three branches of the function defined by
this cubic equation will be of the second order, say
where A and B are functions of :. We have
''+')^+*'(J)"i»
so, substituting in
(».+i)5' + JK+i)f+^(«'+«)o,
and using M'^ + 3;(i=ii, wo have
Multiplying the righthand side by {«j* + 1)^, and the lefthand side by its
equivalent l + wu — vfi, we have
on reduction by the original algebraic equation. This will hold for each of
the three roots of that equation, if
g«'^=«(iJ«.'+K)+^(«^8)l
O^lA'd + 1%" + 'ABu y
These conditions give the values of A and B ; and the equation for w is easily
found to be
dho ( uu u"\dw_ u'^
"^2"'"UH4 u)dz *ii«+4'^
where u' and u" are the first and the second derivatives of u. The equation
oi the seco 1 de as d ted
A 1 When the Igeb a equat on of degree m is of quite
g ner 1 fo m the 1 near 1 fierent al equat n sat sfied by t roots is of order
m B t when the ^gebra c eq at on has orj sj al formi, though still
rre I ble the d fferent al eq at on in y be of order 1 s th m ; for the
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17.] DIFFERENTIAL RESOLVENTS 49
elimination of various powers of w may not require derivatives up to that of
order m. The most conspicuously simple ease is that in which the alge
braic equation is
where K is a rational function of z ; the differentia! equation is
only of the first order.
Other cases occur hereafter, in Chapter v, where quantities connected with
the roots of algebraic equations of degree higher than two satisfy linear
differential equations of the second, order.
^ote 2. The differential equations considered have, in each case, been
homogeceoua. If we admit nonhomogeneous linear differential equations,
viz. those which bave a term independent of w and its derivatives, then in
the general case, where /(w, i) has a term in (t"""*, the differential eqviOtion is
of order m—1 ordy. This can be seen at once from the elimination of
w^, ui*, ..., a;™! between
^ ds ^
leading to a (non homogeneous) linear equation of order m 1. This result
appeal's to have been iirst stated by Cockle*; it is the initial result in the
formal theory of differential resolventst.
&«. 2. Shew that, when the algebraic equation is
the two linear differential equations, homogeneous and nonhomogoneous
respectively, are
^ _ 3t23*i rf«j 3+22=
d^ Z^r^ dz 1'+^'^ '
dm H2s^ #
di s+s^ ^'"' 1+^'
E.V. 3. Obtain the differential equations satisfied by efich root of
(i) ffj3_3^2^^ = 0;
(ii) i!^Zzw^si=0.
Ex. 4. Shew that any root of the equation
rny = {nl)x:
* FUl. Mag., t. sxi (1361), pp. 37fl— 383.
t Foe cefarences, see a paper by Harley, Manch. Lit. and Phil. Memoirs, t. v
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50 SIMPLE ROOTS OF [17.
(n being greater than 2) satisfies tlie eqiiatiuii
wheiw a = l — . What ia the form for « = 2? (Heymann.)
Ek. 5. Shew that any root of the equation
(« lieing greater than 2) satisfies the equation
where the constants a^ arise as the coeffioients in the algebraic equation
_s(iri
jA'
when the roots a
ro
x=(i,)^^'L^
fori = l, .
.., B.
L, and
«..
1=1
Ex.e.
then
Prove that,
if
rV+5y
ix
i^y 2,gl
rT=0;
and explain the decrease in the order of the differential equation.
(Math. Trip., Part ii, 1900.)
^ii^■1)amental system of integrals associated with a
Fundamental Equation.
18. We now proceed to the consideration of the fundamental
equation A = Q appertaining to the singularity a.
The simplest ease is that in which the m roots of that equation
are distinct from one another, say 6^, 9^, ..., 6^ Not all the
minors of the first order vanish for any one of the roots : if they
did vanish, the root would be multiple for the original equation.
Hence each root 6r determines ratios of coefficients c^, c,.j, ..., Crm
uniquely, such that an integral of the equation exists, having the
value
and possessing the property that
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18.] THE FU^DAMENTAL EQUATION 51
where m/ is the value of Ur after z has described a complete simple
contour round a. We thus obtain a set of m integrals.
These m integrals constitute a fundamental system : otherwise
a permanent relation of the form
KlMl + Kstf, + . . . + K
would exist. This quantity Sk^w^ is
=
integral ; as it is zero
and all its derivatives are zero at and near z, it is zero everywhere
when continiied over the regulaj part of the plane. Accordingly,
let z describe a simple closed contour round a: when it has
returned to its initial position, the zerointegral is Sk^m/, that is,
kAHi + icAui +... + K^O,^iim = 0.
Similarly, after a second description of the simple closed contour,
we have
KA'Ur + 'cA^ti, + ... + >c„e„>,^ = 0.
made in this way : we
Let m — 1 descriptions of the contour 1
(i + K
for r = 0, 1, ...
zero, we have
i all the coefficients /
that is, the product of the differences of the roots is zero. This is
impossible when the roots are distinct from one another; hence
the coefficients «,, ..., k^ vanish, and there is no homogeneous
linear relation among the integrals Wi, ..., «,„, which accordingly
constitute a fundamental system.
The general functional character of these integrals is easily
found. Let
so that T^ is a new constant, which is determinate save as to any
additive integer; as the roots 8i, ..., dm are unequal, no two of
the m constants r^, ..., r^ can differ by an integer. Now the
quantity
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52 EFFECT OF A [18.
acquires a factor e "'''', that is, 6^, when z describes the simple
complote circuit round a. Hence the quantity
returns to its initial value after the variable has described the
simple complete circuit round a; and therefore it is a uniform
function of s in the immediate vicinity of a, say tf^, so that
As this holds for each of the integers i^, it follows that we have a
system nf fundamental integrals in the form
where tj>,, 02, ..., (f>m oi'^ uniform functions of z in, the vicinity of
a, the quantities r^ are given by the relations
j„=^log^„,
and the roots 0,, ..., dm of the fandamental equation are supposed
distinct from one another, no one of them being zero.
As regards this result, it must be noted that the functions ^
are merely uniform in the vicinity of a : they are not necessarily
holomorphic there. Each such function can be expressed in the
form of a series of positive and negative pov^ere of z — a, converg
ing in an annular space bounded by two circles having a for a
common centre and enclosing no other singularity of the equation.
There may he no negative powers of ^ — a, in which case the
function >}> is holomorphic at a ; or there may be a limited number
of negative powers, in which case a is a pole of <^ ; or there may
be an unlimited number of negative powers, in which case a is an
essential singularity. Moreover, r^ is only determinate save as to
additive integers : it will, where possible (that is, when a is not an
essential singularity), be rendered determinate hereafter ; so that,
in the meanwhile, the result obtained is chiefly important as
indicating the precise kind of multiform charactei' possessed by
the integrals near a singularity.
19. Now consider the case in which the fundamental equation
A = appertaining to the singularity a has repeated roots, say Xi
roots equal to 0,, X^ roots equal to 8^, and so ou^ where ^,, 6^, ...
are unequal quantities, and Xj + Xj + . . . = m. It will appear that
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19.] MULTIPLK ROOT 53
a gioup of linearly independent integrals is associated with each
such root, the number in the group being equal to the multiplicity
of the root ; that each such group can be arranged in a number of
subgroups, the extent and the number of which are determined
by the elementary divisors connected with the root ; and that the
aggregate of the various groups of integrals, associated with the
respective roots of the fundamental equation, constitutes a funda
mental system.
Group of Integrals associated with a Multiple Root oe
THE Fundamental Equation.
Let K denote any such root of multiplicity a; and let the
elementary divisors of A (8) in its determinantal form be
{««)'—, (e«)"'. .... (««)'"•, (««)'■;
then the minors of order r (and consequently of degree to — t in
the coefficients of ^) are the earliest in increasing order which do
not all vanish when d — k. Consequently, in the set of equations
T of them are linearly dependent upon the rest; hence taking
m — r which are independent, we can express m — t of the con
stants p linearly in terms of the other t, which thus remain
arbitrary. Let the latter be pi, ..., p^; then the integral, given
by
U = p,W, +p2tVi +. + pm1«m.
becomes
u = p,W, + p^W,+ ...+p,W„
where
Tfi = WilA:,+i,W^+i + ... + k„,iW,^
H^2 = Ws + /;,+,, aWr+i + . ■ . + h>i,2^n
W, = iVr + &,+,,TWr+. + ■ ■ ■ + ^m.rW™
and the determinate constants k are given by
Pr+i = h+,^,p, + k,+,^^p^ + ... + k^+j^.
prn = h+ii,ipi + krii,sP^+ ■•■ +k,+2,
Pm =/l^™,iPi +^m,ap2 + ■ +4»,ti
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54 ELEMENTARY [19.
being the expressions for the m — r quantities p in terms of the
T quantities p which remain arbitrary.
Evidently each of the quantities W is an integral of the
equation : and they have thc! property
w;=kW,.
for r = l, ,,., T. Moreover, they are linearly independent; any
non evanescent ieiation of the form
would lead to a relation between w,, ..,,Wm which would be homo
geneous, linear, and nonevanescent, a possibility excluded by the
fact that Wi, ,.., if™ constitute a fiindamental system.
The only case, in which t = o, occurs when the indices cr — o, ,
CTi — (Tj, ..., tTri of the elementary divisors are each unity. In
that case, we have obtained a set of integrals, in number equal to
the multiplicity of the root.
20, We shall therefore assume that t < cr ; and we then use
the integrals W,, ..., W, to modify the original fundamental
system w,, ..., w^, substituting them for Wi, ..., w^. When the
variable z describes a simple closed contour round a, the effect
upon the elements of the modified system is to change them into
IT/, W^', ..., W,', «/',+„ ..., w'n, where
W,f^
cW^,
■...+8.,«^
for r — 1, ..., T, and s = t+ 1, ..., m. The fundamental equation
derived from this system for the singnlaiity a is
.4(n) = o,
where
Ka, ,.
..
,.,.
, «n,.
.
....
0.0,.
., Ka
....
At.... A„....
. A+.„
fi,^
.„+.
n, A+,„,„...
I3,„,,.
/3„ , /3„ ,.
., /3.„
A.
.«
, /3»,,+, ,..,
&„f
.(«n)'A,<ii),
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20.]
where
As « is a root of A (U) of multiplicity <t, it is root of Ai (Xl) of
multiplicity er — T ; and a question arises as to the elementary
divisors of A^ (fi) associated with k.
The elementary divisors of A^ (fl), which are powers of k — Xi,
are
(n«)'.", (n«r'.i, (n «)•—.>, ...
being, in each instance, of index less by unity than those of
A (H). This result, which is due to Casorati*, follows from the
property that .4,(0) is divisible by (li — «)"'; its iirst minora
are divisible by (ii — «)">"''" and not simultaneously by any
higher power; its second minors are divisible by (fi — k)"'"''"^'
and not simultaneously by any higher power ; and so on.
Thi« property, that all the minors of A^ (Si) of order fi ai'e fliviaible by
(jL. 11) '' ' ' and not isimultancously by any higher power, can be proved
a follow t
Aiy mmoi of order ^ of ^ (Q) must contain at least ra—rii of the last
m T columns let it contain mr—ix+a of these columns, where a can
range from to /i. It then miiat contain ro of the first r coJumne.
^im larly it must contain at least mr— it of the last nt — r rows : let it
conta n r /i + a' of these rows, where a' can range from to ^. It then
must cu tai r — q' of the first t rows. The minor may be identically zero :
if not then ow ng to the early columns and early rows that are retained, it is
livisible by (k 2^"", and possibly by a higher power of k — a. Conse
quentlj ome imong these minors are espreasible as the product of {« fi)'"''
by a Imear combination of minora of Ai (Q) which are of order /i ; the coefii
eienta in the combination are composed of the constants, which occur in the
first T — ^ columns and the last mr rows, and thus are independent of O.
But a minor of order it ol A (Q) is not necessarily divisible by a powei' of
K  fi with an index higher than <t ; thus
(i!fi)V. polynomial in fi=(sfi)'""'''.sum of minors of ^, (n).
It therefore follows that the power of «  Si common to all those minors of
A, {a) is of indei not higher than o (t — ^^).
' Comptes Reitdus, t. xcii (I88I). p. 177.
+ Heffter, MMeitmm in die Theorie der Uitearen DiJfsrenUulyleichvngen.
pp. 350—256.
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56 ELEMENTARY DIVISORS OF THE [20.
Kext, we know that thore axo some minora of the original A (ii) of order t,
which do not vanish when JJ = k and which therefore are not divisible by
« — Q. Clearly they cannot contain any of the first t rows in A (Si) ; and thus
they miat ho composed of sets of to — t columnB selected among the last
m  r vows. Take the minors of order )i of any one of these nonvanishing
determinants, their number being N^, where
= ^;
and denote these minora by
ifi (/ A=l, ...,N),
tl togors /adiurrei Iftttl 11 terat on of i et of ^ columns
a d a set f /i rowa o t of the non an sh j, deter ant of order m  t.
Let ft Ije the comj leme tarj of Y,^ n ta w determ na t
Non take tl e nunor of 4 (Q) nh ch ire of o de /t the number ia iV',
a d thej n aj be denoted by o j f o ^ = 1 \ w tl tl e o significance
in the integers as for M/j^. Construct an expression
say, where J^ is a determinant of order mr. Then either (i), J^ vanishes
identically, owing to identities of rows or columns ; or (ii), J^ ia equal to
+ A^(il) and therefore is divisible by (^fl)"'"'', that is, certainly divisible
by («n)°"^ "('■''*, for (§16) wo have
T — (T'l > (T^ — o^ ^ . . . ^ u^ _ 2 5^ 1 ;
or (iii) /ft, when bordered by r — fi of the first rows, and tho first columns in
A (11), is a minor of order ^ oi A (JJ) and ia therefore divisible by (k— !J)''i', so
that the equivalence of tho two espressiona for the minor of A (Q) gives
(k — il)'^f . polynomial in a = (K ilY~''.Jh,
and therefore J^ is divisible by (jtn)V~t''~'''. Tt thus follows that J,, is
divisible by (Kfl)'^» "*''"'*', in every case when it is not aero: and this
holds for all values of /(. Taking then
rai,aii + n(jja(g + ... + mjwai,v=/i,
for /i=i, ..., A' and for one particular value of i, we have a series of A^ linear
eijuations in the quantities a;,, ..., 0;^. The determinant of their coefficients
is a power of the non vanishing detemiinant of order tot, for it ia a
determinant of all its minors of one order : and therefore it doea not vanish.
Hence, so far ispoRe'a t k Jl are conce ed e h of the mnoB a, a v
is a linea co h n t of / ■/> <kll of 1 e e are d s ble by
(kQ)"«" '' d d he ef e h of the m o ■s a. a jf s «rta nly
divisible by h t power The esult 1 olds lo ea<;l t the val es of
It has eea oe t t e i we o k a om no to 11 tl ese n ors of
A^ (Q), has a n 1 ot greater ha a /i omb n^ tl e rea Its, ve
infer that the highest power of k  Q, common to all the minors of A^ l_il) of
order /i, has its index equal to ir  (r  p).
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21.] FUNDAMENTAL EQUATION 57
21, The indices of the elementary divisors of A^ {11) ate
ffo,1, <r,o2l, a,(T,~l, ...;
let there be t' of them, where t'$ t, so that the last t — t' of the
indices of those of A (il) are equal to unity, on account of the
property
o  0, ^ 01  os ^ oa  oa >  . . 3^ T,, > 1.
Then the minors of .^i of order t' (and consequently of degree
TO — T — t' in the coefficients of Ai) are the earliest in successively
increasing order, which do not all vanish when il = k; conse
quently, in the set of equations
Pl'0r.r+, + Pl^r,,^^ + ■ ■■ + pV, /3,,„, = Kp/, (r = r + 1, . , ., to),
t' ol' them are linearly dependent upon the rest. Hence taking
in — T — t' of the equations which are independent, we can express
m — T — T of the constants p' in terms of the other t', which thus
remain arbitrary and which may be taken to be p,', . . ., p'^.
Now take an integral
v = pM+. + ... + p\n.w..>
and substitute for the various coefficients p' in terms of p,', ..., pV
The integral becomes
V = p! TF"„ + p/ ^fj^ +  . . + p', t^lr' ■
where, writing \ — t + t\ we have
Wir = W^+r + k+i,rVlk+i + . .  + Un,rWm,
forr=l, ...,t'; and the determinate constants i are given by
Pr'+s = /j,+B,ip,'+ ... + ^J.+s, r'pV,
for 8 = 1, .... m — X, being the expressions of the constants p' in
terms of ,0,', .... p',.
Clearly ea<;h of the quantities Wa, ^a, ■■■, W^' is an integral
of the equation. Moreover, they are linearly independent of one
another and of W], ..., W,\ for any nonevanescent linear relation
of the form
F,W, + ... + F,W,^F:W,, + ... + fVTf.y =
would lead, after substitution for W^. ..., Wj, W"„, ..., Wi, in
terms of the original fundamental system Wj, ...,Wm, to a non
evanescont homogeneous linear relation among the members of
that system — a possibility that is excluded.
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58 SUBGROUl'S OF [21.
As regards the effect, which is caused upon each of these
newly obtained integrals by the description of a simple contour
round the singularity, we have
Tf,,' = W\^, + k^,.rVj\+, + ... + ln,,y>«'
= lcW,r+ V,.,
where V,. denotes a homogeneous linear combination of TTi, ...,
W,. Now no one of the quantities Vy can be evanescent, nor
can any linear combination of the form
7iFi + ... +7t'F,'
be evanescent : for in the former case, we should have
and in the latter
(7, w„. + . . . + 7,' w„.y = « {y, If „ + , . . + y,, r,.o
As Wir and y,W„ + ... +'y.,'W,r in the respective cases are linearly
independent of W^, ..., W,, we should thus have a new integral
of the same type as Wi, ..., Wr', and then, instead of having some
of the minors of order t in A (H) different from zero when H — k,
all of them of that order would be zero, and we should only be
able to declare that some of order t + 1 are different from zero :
in other words, the number of elementary divisors of A (H) would
be T + 1 instead of t. The quantities V^, ..., V^ are thus linearly
equivalent to t' of the quantities Wj, ..., W,, say to Wi, ..., TT^';
hence constructing the linear combinations of V,, ..., V^i which
are equal to TT,, ,.., Wj' respectively, and denoting by w,,, ...,
Wir' the linear combinations of W^, .... W^^' with the same coefGci
ente as occur in these combinations of V,, ..., V,', we have a set
of t' integrals «?„, .... Wi^, such that
tVir = KWir+ Wr, (r = 1, ..., t').
These integrals are linearly independent of one another, and also
of Wj,..., Wr, before obtained. They constitute the aggregate
of linearly independent integrals of this type ; for if there were
another linearly independent of them, it would imply that A^ (li)
had t' + 1 elementary divisors instead of only t'.
As regards the two sets of integrals already obtained, it may
be noted, (i), that the set TFi, ..., W^ can be linearly combined
among themselves, without affecting the characteristic equation
WJ^kW,.:
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21.] INTEGRALS 59
(ii), that to aach integral of the set Wn, .... Wi,< there may be added
any linear combination of the integrals of the set Wj_, ..., W^,
without affecting the characteristic equation
If the index of each of the elementary divisors of Ai(il) is
unity, then t'—ct^t, so that the number r + r of integrals
obtained is then equal to <t, the multiplicity of the root of
A (Xi) = in question. In every other case, ■/ + t< a:
22. When t' + t < tr, so that t' is less than the degree of
^i(il), we use the integrals w„, ...,w,,/ to modify the funda
mental system W^, ..., W^, w^^j, ...,Wm, substituting them for
w,+,, .... «!,+/ in that system. When the variable z describes a
simple closed contour round a, the effect upon the elements of the
modified fundamental system is to change them into IJV, .... W,',
Wii, ..., w'it', w\^i. ..., wj,, where t ^t —\, and
w; = kW„
w/7ftF,l... + 7„V.
+ 7t,T+iW„ + ... + 7tAWi,' + 7(,A+,WA+,+ ... +7i,mW,„,
for r = 1, ..., t; s = 1, ..., t'; f = \+ 1, ..., m. The fundamental
equation derived through this system is
A (fl) = {«  Iiy+''^, (11) = 0,
where
J.j(Il)=l 7x+,,*+, ft, 7A+i,*+si ■■. 7*+i,™ I
17™,''+! . 7™,^+^ . ■■■, 7m,mft
Also * is of a root of A^in,) of multiplicity <t — tt'. By a
further application of the proposition (§ 20) connecting the
elementary divisors of A (il) and 4, (H), the indices of the
elementary divisors of ^^{n), which are powers of k — CI, are
seen to he
ff — (7] — 2, (7j — (Tj — 2, (T^ " o^s — 2, ....
say t" in number.
The procedure from the equation A^ (fl) = to the corre
sponding subgroup of integrals is similar to that adopted in the
case of the equation .A, (H) — ; and the conclusion is that there
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60 COMPOSITION OF A GROUP [22.
exists a subgroup of r" integrals w^i, iCs,, ..., Wj^", characterised
by the equations
for ( = 1,2, ...,t".
And so on, for the subgroups in succession. Combining these
results, we have the theorem* ;
When a root k of the fmidammtal equation A (fi) = is of
■multiplicity/ <r, and when the elementary divisors of A (12) associated
vjith that root are
a group of a lineai ly mdtpendpnt iiitegt als is associated with that
root : this group constat' if a mimher /r — ai of subgroups, which
satify the equatioii',
Wf = KWr, for r=l, T,
Wa = KW^ + w,t, for t = l, ..., t",
and so on. The integer r is the number of elementary divisors of
A (ii) ; t' is the number of those divisors with an index greater
than unity ; r" is the number of those divisors with an index greater
than two ; and so on.
The group of <r integrals, and in~ a other integrals, all
linearly independent of one another, make up a fundamental
system : tlie m. — a other integrals being associated with the
m — (T roots of ^(li)=0 other than D^—k. When these roots are
taken in turn, wc have a single integral associated with each
simple root, and a group of integrals of the preceding type asso
ciated with each multiple root, the number in the group being
equal to the order of multiplicity of the root. We thus have a
system of integrals of the original differential equation distributed
among the roots of the fun<lamental equation associated with the
" That pait ot the theoiem which establialiei the e^isteiice of the group of
inteitralB ata ttated with a multiple loot is due to Fuoh'i Cielle t lwi (18fl6),
p. lifb but the initial eipiession ti'O 'o 'he members of the group was much
moie complicated The part which arranges the group m sub gioups each with
its own chaioeteristic eijuafion is due to Hamburgei CrdU t Litvt (1873)
p. 121 he takes it in an aiiangement \vhieh will be found in the next seotiou
The association of the subgroupR with the elementaiy divii^oic of i \Q] i& due to
Canorati Compte' Kemiits, t sen (1881) p 177
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22,] OF INTEGRALS 61
singularity : that the system is fundamental is manifest from the
facts, that the initial system was fundamental, and. that all modi
fications introduced have been such as to leave it fundamental.
Ex. 1. Two independent int^rala of the equation
Hence when tho variable describes a sirajilc closed contour round the origin
it) the positive direction, we have
and therefore the fundamental equation belonging to the origin (which is a
singularity of the equation) ia
I \~e, 1=0,
I 2jrt , \^e ■:
that is, it is
(fl+i)'.o.
Similarly, two independent integrals of the equation
„dho , dw „
are given by
Hence aftei' a simple closed contour round the origin, we have
where n is e^" ; the fundamental equation belonging to the origin is
: l_S, 1 = 0,
I , a6 '
thatia,
Ex. 2. Consfruct the linear differential equation of the third order,
having
for three b early ndepe del t ntegra!'' btain the fundamental equation
apperta n ng to the r g as a a gular ty ; and from the form of the
diflere tal eq at nn ver ty P" car^a theorem (§14) that the product of
the three r t's of th a t ndame tal e'j t n ia unity.
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62 hamburger's [23,
Hamburger's Resolution of a Group of Integrals into
SuBoROUpa
23. In the case when the roots of the fundamental equation
are all distinct from one another, the general analytical character
of each of the integrals of the fundamental system in the vicinity
of the singularity has been obtained ( 18). We proceed to the
corresponding investigation of the general analytical character
of the group of integrals in the vicinity of the singularity, when
the group is associated with a multiple root of the fundamental
equation.
We have seen that the group of iiuearly independent integrals
can be arranged in subgroups of the form
W„ W.„ .... W, ;.
the members of eaeh subgroup being aiTanged in a line and
satisfying an equation characteristic of the line. Let these be
rearranged in the form*
1^1, Wn, Wu, w^, ...
Tfa, W]s, Ws3, Wjs, ■■
each of the integrals in the new line satisfies an equation, and the
set of characteristic equations for any line is, in sequence, the
same as for any other line, so far as the members extend. When
any such line is taken in the form
where the integer fi changes from line to line, the set of the
characteristic equations is
* These are Hamburger's subgconps; see note, p. 60. Their number is equal
>o the nwnbei of elemental; diviBois of A (Q) connected with the multiple root.
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23.]
SUBGROUPS
Let
we have
and therefore
[(^
'iiria = log K
»)•]■ = «(.
[% (z — a)~"]' = «! {z — «)~°.
Thus «, {z — «)~" is unaltered by the description of a simple closed
contour round a; it therefore is uniform in the vicinity of a, but
it cannot be declared holomorphic in that vicinity, for a might be
a pole or an essential singalarity of u^ {z — a)"". Denoting this
uniform function of 2 — « by ^i, we have
u, = {zaY^^.
To obtain expressions for the other integrals, Hamburger*
proceeds as follows. Introduce the function L, defined by the
relation
i.ilog(.,.),
then, after the description of a simple contour, we have
L' = L + l.
We consider an expression
F(i)=j'=^.,+(''^)t,.i+('';V"''+
where
\ T ) (^1r)! r!'
and the functions ^i,...,^^ are uniform functions of z — a.
Then if, for all values of n, we take
7/^^, = (zay^APF,
where the symbolical operator A is defined by the relation
^F^F{L + 1)  F{L} = F'F,
we have
= (3  a)''«''+^ (A"F+ A^^'F)
" Crdle, t. LJLXT! (1873), p. 122.
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64 GENERIC FORM OF [23.
holding for all values of n. These are the characteristic equations
of the modified subgroup ; and therefore we can write
with the above notations. This is Hambui'ger's functional form
for the integrals,
24. The integrals ii,, ..,, u^ are a linearly independent set
out of the fundamental system ; and the system will remain
fundamental if Mj, . . . , w^ are replaced by /^ other functions, linearly
independent of one another and linearly equivalent to Mj, ,.., m„.
A modification of this kind, leading to simpler expressions for the
subgroup of integrals, can be obtained. In association with F,
take a series of quantities, defined by the relations
»', = ^„
f^ = 11^5 + 2^.^L + y{r,L'',
i^=«. = >^.+ (''>«,i+(^2>..i^4...
Then we have
AF=auV^_, + a,^v^_^ + a„v^^,+ ... +a,,^_,v,.
where the constants a are n on vanishing numbers, the exa«t
expressions for which are not needed for the present purpose.
Then (e  ayv, is a constant multiple of (ir  ayAi^^F, that is,
of u, ; and it therefore is an integral of the differential equation.
By the last two of the above equations, (s — aYv^ is a linear
combination of (£— a)°A''~'F and (z — ayA''~'F, that is, it is a
linear combination of Jia and Mi ; it therefore is an integral of the
differential equation.
By the last three of the above equations, (2 — ayv^ is a linear
combination of (e  afA'^'F, {z  afA^'^F, {z  a)''A»'^; that is.
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24.] A GROUP OF INTEGRALS 65
it is a linear combinatioa of u,, Us, Mj ; it therefore is an integral
of the differential equation.
Proceeding in this way, we obtain /j. integrals of the form
{^~ay,„ {zaYv, (^a)«,.
Moreover, these are linearly independent, and so are linearly
equivalent to it,, ,.., m^; for, having regard to the expressions of
AF, ..., A«~'f, we see at once that any homogeneous linear rela
tion among the quantities v^, ...,% would imply a homogeneous
linear relation among the quantities F, AF, ..., A''~^F, that is,
among u^, ..., u^; and no such linear relation exists. Hence
Hamburger's subgroup of integrals is equivalent to {and can be
replaced by) the subgroup
{.~o).„ (^.)., (^o)»„
Accordingly, we now can enunciate the following result as
giving the general analytical expression of the group of integrals,
associated with a multiple root « of the fundamental equation*:—
Wke}i a root k of the fundainental equation A{d)—Q is of
multiplicity a, the group of <t iniegrals associated •with that root
can he arranged in avhgroups ; the number of these subgroups is
equal to the number of elementary divisors of A($) which are
powers of k — 6; the number of integrals in any subgroup is
determined by means of the exponents of the elementary divisors;
and a subgroup, which contains fj. integrals, is linearly equivalent
to the /J. quantities
(zay,,. {zafv, (^»).„
where 27ria = log k, and the ji. quantities y are of the form
V, = yjr, + 2^^L + f,L\
' This form of expresHion for the gronp of integrals appears to have been given
first by Jiirgens, CreUe, t. Lxsx (1875). p. 154. See also a memoir by Fuchs,
Berl. SitzangiUr., 1901, pp. 34—48.
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66 SUBGBOUP OF INTEGRALS AND [24.
where L = „ .log(z — a), (^ ] denotes . q r;~, . f"*"^
Stti ^' ' \ r j . {p.\—r)\ t\
the jj. quantities i/^j, ;.., ■^^ are uniform (but not necessarily
holomorphic) functions of z — a in the vicinity of the singularity.
Dll'FEBENTIAL EqL'ATION OV LoWEE ObDEK SATISFIED BY
A SubGroup of Integrals.
25. The preceding form of the integrals in each subgroup of
a group, associated with a multiple root of the fundamental
equation, has been inferred on the supposition that the coefficients
of the linear equation are uniform functions.
It will be noticed that the coefficient of the highest power of
L in each of the members of the subgroup is the same, being an
integral of the equation, — a result which is a special case of a
more general theorem. Moreover, it is of course possible to verify
that each member of the subgroup satisfies the differential
equation ; and it happens that the kind of analysis subsidiary to
this purpose leads to the more genera! theorem above indicated,
as well as to a result of importance which will be useful in the
subsequent discussion of the reducibility of a given equation. We
proceed to establish the following theorem* which is of the nature
of a converse to the theorem just established :
If an expression for a quantity u be given in the form
U = tj}n + <f>n,L + ^„^L^+ ... + ^ai"' + ii>iL"\
where L = ^ — . log (z — a), and each of the quantities is of the
form
(^ = (^ — a)' . uniform function of z~a,
a being a constant, then u satisfies a homogeneous linear differential
equation of order n, the coefficients of which are functions of z
uniform in the vicinity ofz = a; moreover,
du ^ 3"'M
dL' dL" '"' 9i»'
are integrals of the same equation and, taken together with u, they
constitute a fundamental system for the equation.
" Fiiclis. in the memoiv quoted on the pceeedins page.
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25.] ITS DIFFERENTIAL EQUATION 67
(It is clear that „,^^^ is a numerical multiple of ij>,, and that
the coefficient of the highest power of L in each of the announced
integrals is, save as to a numerical constant, the same for all ; it
is a multiple of 0i, which is an integral of the equation.)
It is convenient to make a alight modification in the form of u ;
we take
■■■+("i')+'^""+ *''''"■
where
so that the character of the functions i^ and their form (except
as to a mere numerical constant) are the same as those of the
functions i^. Further, no change, either in the property that
dii_ dhi
are integrals of the equation or in the property that, taken
together with u, they constitute a fundamental system, will be
caused if they are multiplied by constants : so that, if the theorem
can be established for n,, ..., m„_j, where
1 3"'« ,
1 ! 8"i, , , , r
y^,L'^^ + ir,L«'
the theorem holds for the quantities as given in the enunciation
of the theorem.
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68 SET OF LINEARLY [26.
26. Merely in order to abbreviate the analysis, we take m = 4 ;
with the above forms, it will be found that the aaalysis for any
particular caee such as n = 4 is easily amplified into the analysis
for the general case. Accordingly, we deal with quantities w, u,,
ui, u?,, where
M = i^j + S^jL I Z^M + ir^L",
If u can be an integral of a linear equation of the fourth order
with coefficients that are uniform functions oi' s — a in the vicinity
of a, let the equation be
dz' ds^ dz^ ds
Let the variable z describe a simple contour round a ; this leaves
the differential equation {if it exists) unaltered, and so the new
form of u is an integral, say u, where
u'=Kf,+ BKf,{L + 1) + dKf,{L + ly + K^p■,{L + If,
where k is the factor comnjon to all the functions ifr after the
description of the circuit. As u and u' are integrals of a homo
geneous linear equation, so also is
Hence v also is an integral, and it is given by
v' = B {«V^a + 2«i^, (i + 1) + K^i (£ + If}
+ 3 [Kf, + Kf, (L + 1)1 + K^, ■
and therefore
w, = ^ [  I'' — w I = Mj + M, ,
ntegral. Hence w' is also an integral, and it is given
by
and therefore
is also an integral.
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26.] INDEPENDENT INTEGRALS oS*
Thus integrals are given by
t, =■«!,
■wt, =«5,
which proves one part of the theorem, viz, that u, u,, u^, Uj are
simultaneous integrals of the linear equation if it exists.
27. In order to estabhsh the property that w, Ui, u^, % con
stitute a fundamental system of the equation if it exists, a pre
liminaiy lemma will be useful; viz. if A, B, G, D be functions
free from logarithms and if they be such that a simple closed
contour round a restores their initial values, except as to a con
stant factor the same for all, then no identical relation of the
kind
aA + 0BL + yCD + BDL' =
can exist, in which a, /3, y, S are constants different from aero.
For let the simple contour be described any number, N, of times
in succession ; and let / be the constant factor acquired by the
functions A, B, 0, D after a single description of the simple
contour. Then we should have the relation
/"[aA + 0B{L + N) + yC{L + Nf + SD(L + N'f] = 0,
and consequently the relation
aA + 0B(L + N} + yG(L + Ny+?iD{L + Nf^O.
valid for all integer values of JV". Consequently, the coefficients of
the various powers of JV" must vanish : hence
0= SD.
= 38Z)i + yC,
= 3SDi= + 27CZ I &B,
0= hI)L'\ yGL^ + l3BL + <xA,
the last of which is the original postulated relation. From the
first of these relations, it follows that
S =
then, from the second, that
7 =
then, fiom the third, that
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70 EQUATION SATISFIED BY [27.
and so, from the original relation, that
a = 0.
The lemma is thus established.
It may also be proved that, if A, B, C, D be functions free
from iogarithma, and if they be such that a simple closed contour
round a restores their initial values, except as to constant factors
which are not the same for all, then no identical relation of the
kind
aA + ^BL + riCD + BDL^ =
can exist, in which a, 8, 7, 2 are constants different from zero.
The proof is left as an exercise.
It is an immediate inference from the course of the lemma
that no relation of the form
a'u + ^Ma + Jli^ + B'Ui =
can exist, in which a, ^', i, S' are constants different from zero ;
for proceeding as before, it would require
0= a'.r„
= 3a' ■f 3 + 2/3>5 + 7>i,
= n'>^, + ^'^s + iifi + ^'^1,
which clearly are satisfied oniy if a' = ^' = 7' = 8' = 0. Hence there
is no homogeneous linear relation among the quantities m, Mi, Ws,
Ms ; and they therefore constitute a fundamental system for the
linear equation if it exists.
28. If the equation exists, we must have
and in the operator A, the functions P, Q, ii, S are to be uniform
functions of z in the vicinity of a. Let Z denote any function of z
with the same characteristic properties as ^], ^.j, if~a. "^4; then
with such an operator A, we have
A(^i) =iA2 +Z',
A(2'i=) = Z=A.?+2L2' ■\Z",
A (^i') = i^AZ + ZL'Z' + %LZ" + Z"\
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28.] A HET OF IiNITGGRAJ.S 71
where Z' , Z", Z'" are functions of the same characteristic properties
as Z, that is, aa i/^j, i^^, Jcj, i^^, and they are iree from logarithms.
Now as A?i, = 0, we have
A.^.  0.
As A«,; — 0, we have
ii^,j + iA'^i + l^i' = 0,
that is,
As A«a = 0, we have
Af 3 + 2 (LAi^i 4 f /) + i^A^, + 2i^/ + ./r," = 0,
that is, by using the two preceding relations,
As A((=0, wehave
A>/rj + 3 (LAa/t, + ■^;) + S (Z^A»/r, + 2if / + 1/^;')
+ i^Ai/Tj + ZL^; + 3i/f," + ^1" = 0,
that is, by using the throe preceding relations,
Ai^^ 4 3i^/ + 3f /' + i/r,"' = 0.
Thus there are four equations ; each of them involves the coeffici
ents P, Q, R, S linearly and not homogeneously. The required
inferences will be obtained if the equations determine P, Q, R, S
as functions of s, uniform, in the vicinity of a.
Now each of the functions yjr is such that {z — ay'yjr is a
uniform function of 2  a in the vicinity of a ; accordingly, let
(za)^f^=0^, (/t=l, 2,3,4),
where each of the ^'s denotes a uniform function. Substituting
(z — ay$i^ for ^1^ in each of the four equations, the factor (s — a)'
can be removed after the dififerential operations have been per
formed ; and then all the coefficients of P, Q, R, S, and the term
independent of them, are uniform functions of ^ in the vicinity of
a. Solving these four equations of the first degree for P, Q, M, S,
we obtain expressions for them as uniform functions of s — a in
the vicinity of a. (In general, this point is a singularity for each
of the expressions.) It follows that, for these values of P, Q, Jf, S,
the four quantities Wj, u^, v^, u are integrals of the linear differen
tial equation of the fourth order.
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72 PROPERTIES OF A SUBGKOUP [28.
As already remarked, similar analysis leads to the establish
ment of the result for the general case ; and thus the theorem is
proved.
Corollary I. It is ao obvious inference from the preceding
theorem that, when a group of integrals is associated with a
multiple root of the fundamental equation, any (Hamburger)
aubgi'OTip, containing (say) u of the integrals, is a fundamental
system of a linear equation of order n with uniform coefficients.
Further, it is at once inferred that the n' members of that sub
group, which contain the lowest powers of the logarithm, constitute
a fundamental system for a linear equation of order n' with uniforiu
coefficients.
Corollary II. Similarly it may be established that one
(Hamburger) subgroup containing n integrals, and another sub
group containing p integrals, constitute together a fundamental
system for a linear equation of order wl^ with uniform coefficients.
And so on, fur combinations of the subgroups generally.
Jix. Prove that if the linear equation in w has a sabgroup of n iiitegi'als
which, ill the vicinity of a singularity a, have the form
W3 = ./'3 + 21;'^Z;i^/',Z^
where 2ii2X = li>g(am), and oaflh of the functions^ is sueli tbat (sa) "^ is
uniform, where e^'^ is a multiple root of the fundamental equation with
which the subgroup of integrals is associated, then if the linear equation for
V he eonatructed, where
that iineiir equation has a corresponding auhgroup of n
 1 integrals of the
where the functions ip are of tlie same character aa the functions tJt.
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CHAPTER III.
Regular Integrals; Equation having all its Integrals
rbqular near a singularity.
29. The general character of a fundamentai system of
integrals in the vicinity of a singularity has now been ascertained.
For this purpose, the main property of the linear equation which
has been used, is that a is a, singularity of the uniform coefficients ;
the precise nature of the singularity has not entered into the
discussion. On the other hand, the functions <p which occur in
the integrals are merely uniform in the vicinity of a : no know
ledge as to the nature of the point a in relation to these functions
has been derived, so that it might be an ordinary point, or a
pole, or an essentia! singularity. Moreover, the index r in the
expressions for the integrals is not definite ; being equal to
s— ■ log 6, it can have any one of an unlimited number of values
differing from one another by integers. Hence, merely by changing
r into one of the permissible alternatives, the character of a for
the changed functions ^ may be altered, if originally a were
either an ordinary point or a pole : that character would not be
altered, if a originally were an essential singularity.
It is obvious that the character of a for the integral is bound
up with the nature of a as a singularity of the differential
equation, each of them affecting, and possibly determining, the
other. Accordingly, we proceed to the consideration of those
linear equations of order m such that no singularity of the
equation can be an essential singularity of any of the functions
1^, which occur in the expression of the integrals in its vicinity.
In this case, the functions (f>, which are uniform in the vicinity of
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74 REGULAR INTEGRALS [29.
a and therefore, by Laurent's theorem, can be expanded in a series
of pasitive and negative integral powers of ? — a converging within
an annulus round a, will at the utmost contain only a limited
number of negative powers. To render r definite, we absorb all
these negative indices into r by selecting that one among its
values which makes the function ^ in an expression
{zaf,!,
iiuite (but not aero) when e= «.
An integral of the form
if = (3  ay [<^, + <^, log (3  a) + . . . + <^, log (2  a)Y\
where 0^, 0i, ..., 0, are uniform functions having the point a
either an ordinaiy point or a zero, is called* regular near a.
When a value of r is chosen, such that {z — a)~''u is not zero
and (if infinite) is only logarithmicaiiy infinite like
Co + c, log (2 «) + ... +c„ [log (s »)}",
the integral is said to belong to the index (or exponent) r: the
coefficients c being constants and not all of them zero. Similarly,
when the singularity a is at infinity, and there is an integral
^^, + t.'''g^++t(logJ)j.
where ■^„, ifi, ..., i^, arc uniform functions having s = i» for an
ordinary point or a zero, the integral is said to belong to the index
or exponent p.
It will be possible later to consider one class of integrals that
do not answer to this definition of regularity : but it is clear that
regular integrals, as a class, are the simplest class of integrals, and
that the first attempt at obtaining integrals would be directed
towards the regular integrals, if any. Accordingly, we proceed to
consider the characteristics of linear differential equations which
possess regular integrals : and in the first place, we shall consider
equations all of whose integrals are regular in the vicinity of one
of its singularities in order to determine the form of equation in
that n tj
A Th m I ) P T name for a
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AND THEIR EXPONENTS
As subsidiary to the investigation, one or two simple properties, associated
with the indices to which the fuQctioiM helong, will first be proved.
If a regidar fimction u (in the present sense of the term) hdong to an
index r avd another v to an index s, tken Miw belongs to the index r — s: as is
obvious from the definition.
If a regular function u belong to am index r, t/ien j belongs to Ike index
r—^, To prove this, let
J.(.«)—[_M>*.+(.«)*;)<loi5(,,.)}+.*.pog (.»))— ],
SO tiat T can onlj belong to the index r  1, if some at least of the coefficients
of powers of log{2 — w) are different fTOm zero when z = a. These coefficients
when s = a is substituted: they cannot all vanish, for then </)„, ip,, ..., ^
would vanish when s = a, so that o,, c,, ..., i!^ would all be zero, and then u
would not belong to the index r. Thus r belongs to the indes r — 1.
There is one slight exception, viz. when « is uniform and the index r is
zero ; then y is also uniform, and it may even vanish when 2 = a ; so that,
if •« were said to beloi^ to the index 0, 5 could be said to helong to an index
not less than 0.
Form of the Differential Equation when all the
Integrals are regular near a Singularity.
30. As a first step towards the determination of the form of
a differential equation that has all its integrals regular, we shall
obtain the index to which the determinant of a fundamental
system belongs. Let the system be w,, w^, ..., w„: and let the
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76
Ft
>RM OF DIFFERENTIAL
EQUATION
[30.
indices of the members be r„ r^
the determinant in the form
...., r
■,„ respectively.
We take
Cw/^tf,'"'
'«,".
ofS
3^2,
where G i
s a constant.
The
: quantity
% is a solution
of an
equation, a fundamental
system for which is given by
It is clear that, if w,, w^, ... are all free from logarithms, then
%, Vs, ... are also free from them. If hovrever there bo a group or
a subgroup of integrals associated with a repeated root of the
fundamental equation, we may take (§ 23)
Wi' = Ow,, w/ = Wi + 6w^,
so that
, _ d /w.J\ _ ^ A ^\
' dz \W7 dz\8 wj "
thus 1), is uniform and therefore free from logarithms. Similarly,
Ui and all the quantities used in the special form of the determ
inant are free from logarithms.
The indices to which Vi, %, ... respectively belong are
ra — r, — 1, r, — ri ~ 1, r, — r^ — 1, . . .
unless it should happen that, for instance, r2 = rj. In that case,
we replace Wa by w^ + av^i, choosing a so as to make the new
integral belong to an index higher tlian r^ or r, : this change will
be supposed made in each case where it is required.
Again, the quantity m, is a solution of an equation, a funda
mental system for which is given by
' dz \Vi' ' ' ds \vj '
The index to which % belongs is
r,r,l(r,r,l)l, =..r.l,
and so for u^, ... ; that is, their indices are
'*a — ^a — !> r^ — r^—l,....
And so on, down the series.
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30.] HAVING REGULAR INTEGRALS
Hence the index to which
= w, + (wil)(r,r,l) + (m2){r,r,l) + .,.
... + l(r.r..l)
so that, denoting the determinant of the fundamental system by
A {z) as in § 9, it follows that, in the vicinity of the singularity a,
we have
A (s) = (2  a)''+^=++^'»!™'"'^i a (s  a),
where Ji is a holomorphic function of its argument in that
immediate vicinity, and does not vanish at a.
31. This result enables us to infer the form of the differential
equation in the vicinity of the singularity a. Manifestly, the
equation is
d'^w d'"'w d™~^w
where A is the determinant of the m integrals in the fundamental
system, and A, is the determinant that is obtained from A on
d'^'Wg . ^, , d™w»
. , , for s = 1, . . . , in, by the column , — ,
dz"'" •' dz™
fors = l, ..., m.
Now consider a simple closed path round a. After it has been
described, A and A^ resume their initial values multiplied by the
same constant factor, which is the nonvanishing determinant of
the coefficients a (§ 13) in the expressions for the transformed
integrals ; thus p^ is uniform for the circuit. Hence, when the
expressions for the regular integrals are substituted in A and A„,
all the terms involving powers of log {z — a) disappear. Moreover,
A belongs to the index
n+...+r™Jm(ml);
and so far as concerns the index to which A^ belongs, it contains a
column of derivatives of order k, =m — {i)i — k), higher than the
corresponding column in A, so that A„ belongs to the index
n + ...+r,.im(ml)«.
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78 METHOD OF FEOBENIUS [31.
Hence p^ belongs to the index — k and therefore, in the immediate
vicinity of a. the form of j), is given by
where, at a and in tiie immediate vicinity of a, the function
Pk{z — a) is a holomorphic function which, in the most general
instance, does not vanish when z = a, though it may do so in
special instances. As this result holds for k = 1, ..., m, we con
elude that, when a homogeneous linear differential equation of
order m has all its integrals regular in the vicinity of a singularity
a, ike equation is of the form,
in that vicinity, where P,, Pj, ..., P^ are holomorphic functions of
z — ain a region round a that encloses no other singularity of the
equation.
Construction of Regular Integrals, ey the Method
OF Fbobenius.
32. The argument establishing this result, which is due to
Fuchs*, is somewhat general, being directed mainly to the
deduction of the uniform meromorphic character of the coefficients
of the derivatives of w in the equation. No account is taken of
the constants in the integrals i and it is conceivable that they
might require the esistence of relations among the constants in
the functions P,, ..., P,„, Hence for this reason alone, even if for
no other, the converse of the above proposition cannot be assumed
without an independent investigation. The conditions, which
have been shewn to apply to the form of the equation, are
necessary for the converse: their sufficiency has to be discussed.
Accordingly, we now consider the integrals of the equation in
the vicinity of the singularity!.
Denoting the singularity by a, we write
za = x. PAza) = pr{x)=p., (r = l, ...,m);
* CrelU, t. Lsvt (1866), p, 146.
t The following method ia due to Frobciiius ; leferenMS will be given later.
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32.] FOR REGULAR INTEGRALS 79
SO ihat the equation can be taken in the form
valiii in the vicinity of a: = 0.
If regular integrals exist in this vicinity, they are of the form
indicated in §§ 18, 24, the simplest of them being of the form
wf = a^ 2 g„x° = 2 g^xi''^"
say ; should this be an integral, it must satisfy tlie equation
identically. We have
D«= ,7(<71)..,(» m+ !)»(„ !)...(» m+2)j),.. .
say. Here,y(i«, tr) is a holomorphio function of x in the vicinity of
the iCorigin and is a polynomial of degree m in (j, the coefficient of
(7™ being unity i so that, if it be arranged as a powerseries in x,
we have
f(x.,).f,{„) + ^fA,) + xM,) + ....
where /„ (tr) is a polynomial in ff of degree m, and /,{o), f^ (/r), . . .
are polynomials in cr of degree not higher than m—1. Then
= !/.«'*'/('» P + »)
= iai'+fj,/.(p + ») + y,_,/,(p + «l) + ... + ?./,«).
If the postulated expression for w is to satisfy the equation, the
coetBcienta of the various powers of x on the righthand side must
vanish : hence
oi/./.W,
»/.((>) +?./.(p + i),
OS./.W + »/,((> + 1)+S./.(P + 2),
and so on. These equations shew that the values of p, which arc
to be considered, are the roots of the algebraical equation
fMo
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80 CONSTRUCTION OF [32.
of degree m in p: and that, for each such value of ^,
s. = ,
_!h_
7.(P + l)/.(P + 2)'/.(P + ») ""'
vhere — h^ (p) is the value of the determinant
/.(p + i). , , Me) .
/.(p + i), /.(p + 2), ,., , /.(p)
/.(p + i). />(>>+2). /.(p+8) , Mf)
/„(, + !), /..(p + 2), /„(p + 3) f,{p + ,\), /,_,(p)
/„((.+ !), /..(p + 2). /..<P + 3) /,((. + "!). /.W
SO that h, (p) is a polynomial in p.
If no two of the roots of the equation /, {p) — differ by whole
nnmbers, then no denominator in the expressions for the si
coefficients (j„ vanishes ; the expression £r(«, p) is formally a
for an integral, but the convergence of the series must 1;
lished to ensure the significance of the e
If a group of roots of the equation /„ (p) = differ among one
another by whole numbers, let them be
P. p + e^. , p + e,
where the real part of p i< the smallest, and that oi p + e is the
largest, among the real pait** of these roots; equality of roots
would be indicated by coiiesponding equalities among the positive
integers 0, e^, ..., e. We then t^ke
g,Mf + i)...f.(.P + i)s,
and thus secure that no one of the coefficients g„ becomes infinite.
The condition, that the equation shall formally be satisfied, has
imposed no limitation upon ^j, which accordingly can be regarded
a'i arbitrary ■ hence tj also can be regarded as arbitrary.
33 I o ie to Jc 1 v th both sets of cases simultaneously,
tl e 1 il exp es n s on t ucted in a slightly different manner.
A ] a n etr c quant ty k s ntroduced and it is made to vary
wt!n eg s round the oots of /o(p) = 0, each such region round
a ro t be ng chose o is to c itain no other root. The quantity
g n tl e hr t Set of c s and the quantity g in the second set,
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33.] REGULAR INTEGBALS 81
are arbitrary ; they are made arbitrary functions of a. Quantities
rji, t^i, ... are determined by the equations
»,/.(« + !) + <;,/, (a),
= ft/.(» + 2) + !7,/,(»+l) + <7./.W,
the same in form as the earlier equations other than the first :
these quantities g are functions of a. Moreover, we have
''■<''>' /.(»M)/.(»'l1).../.(.+ .;) ''<°>^
in consequence of the assumption as to g^ (a) in the second set of
cases, and of the regions round the roots of /„(/)) = in which a
varies, it follows that the quantities gi, g^, ... are each of them
finite for all variations of a within the regions indicated. We
thus have an expression
y=g{x,a)^ 2 g.x+'',
also
= i^£r.x"+" /<*.« + ")
= ^„(«)/„(»)^,
the coefficient of every power of x except x" vanishing, in conse
quence of the law of formation of the quantities g.
34. We proceed next to consider the convergence of the
powerseries for y, before bringing the equation satisfied by y into
relation with the original differential equation. We denote by R
the radius of a circle round the a;origin within and upon which
the functions pi, .... pm are holomorphic : so that the circle lies
within the domain of this origin. Then/(ic, a) and its derivatives
with regard to x are also holomorphic for values of x within the
circle and for all values of a considered. As the first of them, say
/' (x, a), is of degree in a. one less than f{a:, a), it is convenient
to consider that first derivative : let M (a) be the greatest value
oi\f' {x, a) for the values of a; and a, so that, as
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82 CONVEKGENCE OF THE [34.
we have*
and therefore, as k + 1 is a positive integer ^ 1, also
i/„<»)«Bif(«).
By the definition of the regions of variation of a and the signi
ficance of the integer e, it follows that the quantity /o(a + j^ +1) is
distinct from zero, for all values of w ^ e and for all values of a ;
hence, as
from the equations that define the coefficients g, it follows that
< i/^(.^,^ + i)  li.!;.lS'*(«)+ L?.B+«(" + 1) + ...
say, where 7^+1 denotes the expression on the righthand side.
Evidently
7.H.,i/.(a + »+ 1)17. !/.(«+ >')J! = l!;.«"<« + »)
« 7, i¥(a + »),
and therefore
Let a series of quantities F^ be determined by the equation
jf(.+») 1 1 /.(«+^) 11
ii/.(a+rti)rj2/.(«+»+i)Ir
for values of I'^e; and let r, = 7,. Then all the quantities F
thus determined are positive, and we have
ls'.«l<7.t.<r.„.
Consider the series
r,if + r,+,a;+>+... + r,a! + ...;
r.+,rJ
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34.] SERIES FOR THE INTEGRAL 83
its radius of convergence is detennined* as the reciprocfxl of
Lim ^ ,
Now M{d) is the greatest value of the moduhis of
 {0  1)...(0  m + 2) p,'  ...  pr^'
within the circle ]aT = 7i. As the functions p/, ...,p,„' Eire holo
morphic within the circle, there are finite upper limits to the
values of [pi'l, ..., \pm'\ within the region, say M^, ..., M^; then
say, where ^ = a. Again
f,(e)=0(di)...(6m + r)$(e~r)...(0m + 2)pAo)...
...P^m,
ao that, if
^(»)„" + ,r(, + l)...(» + ml)
+ ,(,7 + l)...(,r + m~2)!p,(0)+... + lp,(0),
we have
/.(fl)>i9 !/.(«) 91 St" !/.(«) 91,
and
the term in 0™ being absent from /, (0)  0'^, and the term in ff™
being absent from i^ (a). Moreover, as these quantities are
required for a limit when v tends to infinity, the quantities a
and will be large where they occur; thus o™ is greater than
(ff), which is a polynomial in a only of degree m — 1. Hence
j/. (9)1 S »*(«).
Returning now to the expression for r^^^ ^ F^ , let /3 denote
al; tlien
c< + ,+ lS>, + l/3,
so tiiat
a + i.+ lS(v+l(3r.
Again,
\a + v+l\tii + X+l3,
so tliat
*(« + i + l)«*(» + l + /3),
and therefore
j/.(» + « + l)>(« + l»*(..+ l+/3),
* Chrjstal'e Algebra, vol. ii, p. 150.
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84 CONVERGENCE OF THE SEEtES [34.
finally, \a + v\^v + 0, and therefore
so that
M{a+ v) ^{" + 0)
\f,(_ci + v+l)\^iv+lm''<l>l^ + l + &y
Now 1^ (rr) is a polynomial in <r of degree m — 1, as also is <j) (o) ;
hence, owing to the term (v + 1 — 0)"^ in the denominator on the
righthand side, we have
«(«+» ) _
iiri/.(«+»+i)r ■
for all values of /9, that is, for all values of a within its regions of
variation. Again, as /„ (a) is a polynomial in a of degree m, it
follows that
.../.(<. + « + !) ■
for all the values of a, and therefore
Using these results, we have
T . r,„ 1
and therefore the series
converges within the circle ja; = B and for the values of a : conse
quently also the series
7.ic' + 7s+i«'+^l ...
converges for the same ranges of variation for x and a. The
addition of a limited number of terms that are finite does not
affect the convergence : and therefore
converges, for values of x within the circle \a:\ — R, and for values
of a within its regions of variation.
Let any region for a be defined by the condition \a~p\^r.
Then the series converges absolutely within the a;circle of radius
It and the acircle of radius r. Let R'<R, and r^<r; and let
K, K denote any finite positive quantities which may be taken
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34.] INDICIAL EQUATION 85
small ; then* the series converges uniformly for values of x and a
siicli that
\M<«.. ap;<r'..
Thus the scries converges uniformly in the vicinity of the aiorigin,
for all values of a in the regions assigned to that parametric
variable.
By a theorem due to Weierstrassf, the uniform convergence
of the series, which is a powerseries in x and a functionseries in
a, permits it to be differentiated with regard to a; and the
derivatives of the series are the derivatives of the function
represented by the aeries within the aregions considered.
Significance of the Indigial Equation.
35. We now associate the factor x° with the preceding series,
and then we have
g (*■, «) = a; S g^a:' = 2 «/.^+'
as a series, which converges uniformly within a finite region
round the ^origin and can be differentiated with regard to a
term by term. (It may happen that the origin must be excluded
from the region of continuity of g (ic, a), as would be the case if
the real part of a were negative ; the origin must then be excluded
from the region of continuity of the derivatives with regard to a,
owing to the presence of terms such as ^„a;°iog3:.)
The function g{a), a) thus determined has been shewn to
satisfy the equation
iij,(=.,<.) = i/.Wi,.(.).
As associated with the original differential equation, this result
requires the consideration of the algebraical equation {hereafter
called the indicial equation)
/.((■)=<>
of degree tn. The preceding analysis indicates that two cases
have to be discussed, siccording as a root does not, or does, belong
to a group the members of which differ from one another by
* Tlie uniform convei^ence with r^ard to a; is known, T. F.,% li,finn. The
uniform Gonvergenoe with regard to a is eatabliahed bj means of a theorem due to
Osgood, Ball. Anter. Math. Soc, t. iir (1897), p. 73 ; see the Note, p. 122, at the
end. of this chapter.
t Ges. Werhc, t. ii, p. 208; see T. F,. i%S2, 33.
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86 INTEGRALS ASSOCIATED WITH [35,
whole numbers (including a difference by zero, so as to take
account of equal roots).
Firstly, let p be a simple root ot fo(p) = 0, in the sense that it
is not equal to any other root and that the difference between p
and any other root is not a whole number. Then when we take
a =p, all the coefficients 3i, ^5, ... in 5 (a;, p) are finite ; we have
that is,
is an integral of the differential equation : it is associated with
the simple root p of the equation fo(p)=0, and it is a regular
integral.
36. Secondly, let />„, p,, ..., pn constitute a group of roots of
/o (p) = 0, differing from one another by whole numbers and from
each of the other roots by quantities that are not whole numbers ;
and let them be arranged so that the real parts of the successive
roots decrease : thus the real part of p^ is the greatest and that of
p„ is the least in the group. In order to secure the finiteness of
the coefficients^,, ^2, ..., it now is necessaiy to take
S.(a)/.(a + l)/.(. + 2) .../.(a + e)j(a), =/(«)<,(.),
say, where e^p„ — pn, and ^(a) is an arbitrary function of a: and
now
Ds, (»,«) I'S («) n /, (« + ,.) rj (a) F(„),
where
j'W^n !/.(« + .)).
Further, there may be equalities among the roots in the group r
let pu, Pi, pj, pic, ... be the distinct roots taken from the succession
in the group as they occur, so that po is a root of multiplicity i,
Pi of multiplicity j  i, pj of multiplicity k —j, and so on. Then in
^(a), there is a factor {a — p^y through its occurrence in /„(«);
there is a factor {a — pi)', through the occurrence of (a — piV"^ in
/o(a), and the occurrence of (a — p,)' in /(,(a + po — pi); there is a
factor (a — Pj)*, through the occurrence of (a — pj)*~J iny„(a), the
occurrence of (a — pj)^ in/o{a + pj — pj), and the occurrence of
(a  pjY in /„ (a + p, pj). Now
0<i<j<k<...,
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36.] A GROUP OP BOOTS 87
so that, for F(a.), p„ is a root of multiplicity i, that is, 1 <it least :
pt is a root of multiplicity j, that is, i + 1 at least ; pj is a root of
multiplicity k, that is, j+ I at least ; and so on. Hence if p, be a
root in the group as arranged, it is a root of F(a) of multiplicity
K + 1 at least ; and therefore
r«>)i .„_
L 3»' J. p.
for /i=0, 1 K. But
Be, (.«,«)=»!<, (.)*■<«),
and g{3:, a) can be differentiated with regard to a; hence
= 0,
for /li = 0, 1 , . . . , K certainly, and for all other integer values of /x.
less than the multiplicity of p„ as a root of ^(a)=0. Conse
quently, the expression
^ r s^g(^,« )"
9p/
say, for the same values of fi, provides a set of integrals of the
equation.
Moreover, each of the distinct roots in the group thus provides
a set of integrals; we must therefore enquire how many of the
integrals out of this aggregate are linearly independent.
37. We first consider the members of any set ; they are
furnished by
for a value p assigned to a, and for a number of values of fi, say
0,1, ..., K. Kow
and therefore
 + ...+(loga;)'Ssr,(a)x',
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88 SUCCESSIVE SETS ASSOCIATED WITH [37.
where it will be noticed that the coefficient of the highest power
of logic on the righthand side is g {x, a). Hence the set is
y, ^wlogx + w,,
1/2= w (log xy + 2^1 log x + w^,
+ icw^i\ogx + w/,,
where the coefficients Wp are independent of logarithms. From
the fact that y^ contains a power of log a: higher than any
occurring in yo,yi, •■•, ^pi it follows (by the lemma in § 27)
that no linear relation of the form
c,]/, + c,y, + ... + c^y^ =
cao subsist among the integrals.
38. Nest, we consider the sets in turn, associated with the
values pn, Pi, pj, ... of a, as airanged in decreasing order of real
parts. The earhest of thorn is given by a = p^ : and it contains
the i members
for /i = 0, 1 i1. Now
/.(«).(« p.)' («  Pi)'' (■< ~ ft)" ■■■(" Pi)"*' A .
*(.).(«?.)'(«?<)* (.tnf ...(»p,)"« A,.
and therefore
/<" lS) " '°  ''•>' <° ^ !'''>' ■ ■ ■ <° ^ l"^'^
where A,, A,, A^ are quantities which neither vanish nor become
infinite for any of the values p„, pi, ..., pi of a Also
g,(„).g(«}f{a).
where g{a) is an arbitrary function of a; so that (/o(a) does not
vanish for a = pn ; a^id therefore the various quantities derived
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38.] ROOTS IN A GROUP 89
from ^o(al for 'x = p„, including ^„{a) itself, given by 'l' ^'^^
fi = 0, 1, ...,1—1 do not all vanish. Further
L 3' Jf.
which is one of the integrals ; as the quantities
do not all vanish, this integral belongs to the index p, ; and the
coefficient of the highest power of log a: is g(x, po) The iirst set
thus gives i linearly independent integrals obtained by taking
fi — 0, 1, ..., i— 1 in the preceding expression. That which arises
from /I = is
where all the coefficients are finite : thus it is a constant multiple
of
y,*. + aA+i A, (p,) + ^'+^^2 (p„) + . . . ,
an integral that is uniquely determinate.
Now consider the second set : it is given by a = pi, and it
contains the members
for^ = 0. ...,il, i,i+l, ...,jl. The value of5f(a^, a)is
As regards the first part of this expression, we note that all
the coefficients gy(a.) for v=0, 1, ..., pa — pi— 1 contain the factor
(a  pi)* ; and therefore all the derivatives
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90 FORM OF THE INTEGRALS [38.
for /t= 0, 1, ..,, i— 1, vanish when a is made equal to pi, while
they do not necessarily vanish for higher values of fi.
As regards the second part of the expression for g («, a), we
wiitc it in the full form
when a=pi, this becomes
which accordingly is an integral, and it belongs to the index p„,
being free from logarithms. But it has been seen that the
integral, which belongs to the index p^ and is free from logarithms,
is uniquely determinate, being g (x, p„) ; hence the foregoing
integral, being the nonvanishing part of g {x, a) when a is p,, is a
constant multiple of (/{x, po), say Kg{a;, p„). It might happen
that K=Q.
A similar result holds for the derivatives of g(a;, a), for the
values /J. = l, ..., t — 1.
Consequently, it follows that the integrals
for )i — 0, 1, ..., *— 1, can be compounded from the integrals of
the first set ; they are i in number, but they provide no integrals
additional to those in the first set ; and thei'efore, without limiting
the range of their own set, they can be replaced by the i integrals
of that set. As for the remainder arising from other values of fi,
they are
^ r I ^yiM a^+^(og^) i ^"1^'^+. . .+(log*y i g^ (p,) wA
Lv=o 9^^ ..=0 o/'i* .=0 J
for /i = i, V + 1, ,7 — 1 Now
».(«) = » (•>)<«?<)'<" ft)' ■■■ {"eifA,.
30 that the quantities
L 3» J...;
for the values s = 0, 1, ..., /t in any one integral, and for the valuea
fj, = i,i + l, ...,j — l in the different integrals, do not all vanish.
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38.] IN THE SUCCESSIVE SETS 91
All these integrals therefore belong to the index pi, and they are
j — i in number. Moreover, the original set of j integrals, com
posed of these j — i and of the replaced i integrals, was a set of
linearly independent members ; and therefore we now have j — i
integrals, linearly independent of one another and of the former
set of i integrals. Thus our second set provides j — i new
integrals, distinct from those of the first set; and each of them
belongs to the index p;. The first of them is given by fi — i:
L.=o 9pi* * .=0 dpi'' J
which certainly contains terms not involving log w; ii j — l>i,
the second of them is
' y g'^^'g^ (pi) „
Ki + l)log.i^?fc) . + ...],
which certainly contains terms multiplying the first power of
log a;; if _^' — 1 > i + 1, the third of them certainly contains terms
involving the second power of log x ; and so on.
The third set among our integrals is connected with the value
a = pj, and it is given by
for fi^O, 1, ...,kl. Now
The coefficients g, («) contain (a — p^' as factor for all integers v
which are less than pi—pj', hence the quantities
vanish for /i = 0, 1, ..., j — 1, and are different from zero only for
fi=j, j + l, .... k — 1, As in the case of the preceding set, the
quantities
\_dai \
.(")
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92 FORM OF THE INTEGRALS [38,
for /i = 0, 1, ..., i — 1, are linearly expressible in terms of the i
integrals of the first set ; while for fi = i, i + 1, ..., j — 1, they are
linearly expressible in terms of the j^i integrals of the second
set, subject to additive linear combinations of the first set. Thus
the integrals in the present set which are given by
r3'9<^.j.)i
for ^ = 0, 1, ..., ^ — 1, provide no integrals linearly independent
of the i integrals of the first set and the j —i integrals of the
second set ; the j integrals in this new aggregate are linearly
expressible in terms of those in the old. Now the present set of
integrals, for ;u.= 0,l, ..., j—l, j, j+1, ..., fc— 1, are linearly
independent of one another; and therefore the integrals for
/^=j' j+ 1. ■■■ ^'1
are linearly independent of one another, of the i integrals of the
first set, and the j —i integrals of the second set. Thus the third
set provides k~j new independent integrals, given by the k — j
highest values of fi. The first of them, determined by f^ = j, is
which certainly contains terms not involving log a:; if A — 1 >j,
the second of them, determined by f=j+ 1, is
*ft r S
8!^(p)^ + 0+l)logaM.
which certainly contains terms multiplying the first power of
loga;; if ft— 1 >^' + 1, the third of them certainly contains terms
multiplying the second power of log te ; and so on. Moreover, it
is clear that all these k~j integrals belong to the index pj.
The law of the successive sets is now clear. The last of thena,
determined by Oi = pi, contains the integrals
^^l
for /i = f, i + 1, ..., n, which are linearly independent of one
another and of all the integrals of the preceding sets already
retained. All these integrals, being h + 1 — J, in number, belong
to the index pi.
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38.] GENERAL THEOREM 93
The results thus obtained maj be summarised as follows:
When the equation fu{p) = hat, a group of roots p„, pi, ..., pn,
which differ from one anothei hy integers (including zero) and
differ from all the other toots by quariiities that are not integers ;
when also the distinct roots aie arranged in decreasing succession of
real parts, so that p^ is a root of TnuUiplicity i, pi is a root of
m/idtiplicity j~i, pjis a root of niultiphoity k —j, and so on, where
pi' pi' pj' ■■■ ***"* distinct from one another and are arranged in
decreasing succession of real parts; then, corresponding to that
group of roots, there exists a group of n + 1 linearly independent
integrals which are regular in the vicinity of the singularity. This
group of integrals is composed of a set of i integrals, which are
given by
r3;:s_(«^
F^l.
for fi = 0, 1, ..., i — 1, and belong to the index p„; of a set of j —
i, which are given by
for iJ = i, i + I, ., ? — 1, and belong to the hidea; pi; of a set of
k—j integrals, which are given by
for fi =j, i + 1, ..., k—1, and belong to the index pj ; and so on,
the last set being composed of n + 1 — I integrals, which are
given by
I 3«' J„,/
for fj. = l, I + 1, ..., n, and belong to the index p;.
is in substantial agreement with that which
A different proof is given by iFuchaf: briefly stated, it amounts to the
establishmont of an integral w,, belonging to the index p,,, to the transforma
tion of the equation of order m by the substitution
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94 INDICIAL EQUATION [38.
into a linear equation in v of order m 1, and to the discussion of this new
equation in a manner similar to that in which the equation of order m is
discussed. Estiositions of the method devised l>y Fuchs will also be found in
memoirs by Tannery * and Fabry t.
39. Ai! the integrals of the differential equation, which has
the specified form in the vicinity of the singularity, are regular in
that vicinity ; their particular characteristics are governed by the
roots of the equation /„ (p) = 0, that is,
p(pl)..,(pm + l)p(pl)...(p~m + 2)p,(0)...p,(0)=0,
the differential equation in the vicinity of the singularity being of
the form
ax'" ,.„i ^ ^ ' dx"^^
This algebraic equation is of degree m, equal to the order of the
differential equation; it is calted the indicial equation of the
smgularity, and the function f{a;, p), of which fa{p) is the term
independent of x. is called the indidal function. From the form
of the integrals which belong to the roots p of the indicial equa
tion of a singularity, and those which belong to the roots $ of the
(§ 13) fundamental equation of the same singularity, it is clear
that the roots of the two equations can be associated in pairs such
that
When the roots of the indicial equation are such that no two of
them differ by an integer, the roots of the fundamental equation
are different from one another ; there is a system of m regular
integrals, and the m members belong to the m different values of
p. When the indicial equation possesses a group of n roots which
differ from one another by integers (including zero), the corre
sponding root of the fundamental equation is of multiplicity n :
there is a corresponding group of n regular integrals, the ex
pressions of the members of which in the vicinity of the singularity
may (but do not necessarily) involve integer powers of log x.
When a root of the indicial equation occurs in multiplicity «,
* Ann. de 1':Ec. Norm., 2' 86v. t. iv (1875). pp, 113162.
+ Thflse, Faculty des Sciences, Paris (188S).
t Cajley, Coll. Math. Paperi, vol. xil, p. 398. The names adopted by Fuchs
are determinirende Ftindameittalgleichung, anAdeterminirende Fuiictton, respectively.
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,39.] LINEAR INDEPENDENCE OF THE INTEGRALS 95
SO that the corresponding root of the fundamental equation occurs
in a,t least multiplicity k, there is a set of k associated integrals,
the. expressions of all but one of which certainly involve integer
powers of log a;.
40. Having now obtained the form of integral or integrals
associated with a root of the indicial equation f^ (p) = 0, we must
shew that the aggregate of the integrals obtained in association
with all the roots constitutes a fundamental system.
First, suppose that the roots of the indicial equation are such
that no two of them differ by an integer; denoting them by
^1, pi, ..., pm, and the m integrals associated with these roots
respectively by wfj, ..., w^, we have
where Pj (s — a) is a holomorphic function that does not vanish
when z = tt. No homogeneous linear relation can exist among
these integrals : for, otherwise, we should have some equation of
the kind
Writing
^, = e2Tip,^ (6 = l, 2, ...,m),
so that no two of the quantities S,, ..., B^are equal to one another,
we can, as in § "18, deduce the equation
c,i?/w, + cA'tVii + . . . + c^Or/w^ = 0,
for any number of integer values of r, from the above equation, by
making z describe r times a simple contour round a. Taking the
latter equation for 7"= 1, ,.., to — 1, the set of m equations can
exist with values of c,, .,., o™ differing from simultaneous zeros,
only if
' , 1 , ..., 1 =0,
which cannot hold as no two of the quantities are equal.
Hence we must have e, = = Cs= ... =Cm, and no homogeneous
linear relation exists : the system of integrals is a fundamental
system.
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96 LINEAR INDEPENDENCE OF [40.
Next, suppose that the roots of the indicial equation caci be
arranged in seta, such that the members contained in each set
differ from one another by integers. With each such set of rcots
a group of integrals is associated, the number of integrals in the
group being the same as the number of roots in the set.
It is impossible that any homogeneous linear relation among
the members of a group can exist: if it could, it would have
the form
If Wi, ..,, Wn involve logarithms, then ( 27) the aggregate coeffi
cient of the highest power of log (z — a) must vanish ; in the case
of each integral in which the logarithm occurs, this coefficient
(§ 25) is itself an integral of the equation, and therefore wo should
have a relation of the form
where the quantities w^, ..., w^ belong to different indices, say
p,., ..., ps, nw two of which are the same; and w^, ..., w„ are free
from logarithms. Dividing by {z — df>, we should have an equa
tion of the form
hAzay'~'''Pr(^  «) + ■■■ + h>P, (^  a) = 0,
where Pr, ■., Ps ^■'e holomorphic functions oiz — a, not vanishing
when z = (i. No one of the indices py — p^\& zero ; no two are the
same : and so the preceding equation can be satisfied identically,
only \ibr = ■■■ = hs. We therefore remove the corresponding terms
from
h{Wi + , . . 1 h,nW„ = 0,
and proceed as before : we ultimately obtain zero as the only
: value of each of the coefficients b.
If Wi, ...,Wm do not involve logarithms, the argument, above
applied to Wr, ., w,, can be repeated: there is no linear relation.
The initial statement is thus established.
If the tale of the groups, the members of each of which
are linearly independent among themselves, is not made up of
linearly independent integrals, then an equation of the form
c,w,...lc™w„ =
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40.] THE INTEGRALS 97
exists. Equating to zero (§ 27) the aggregate coefficient ol' the
highest power of logs that occurs, we have, as above, a relation of
the form
C^Wr + . . . + C,W, + CjilUp + . . . + C5W, + . . . = 0,
where w^, ..., w, belong to one group, Wj,, ..., w^ belong to another
group, and so on. Writing
CrW,4.. +CsW,= F, CpWfp+ ... +C5W,, = Ifj, ...
we have
Tf, + Tr,+ ...=0.
Now let 6i, =6^'"'', be the factor which, after description of a loop
round a, should be associated with W,; let 6^ be the corresponding
factor for W^; and so on: the quantities 0i, &^, ... being unequal
to one another, because TT,, W^, ... belong to different groups.
Then, as in ^ 18, we deduce the equation
^,*ri + i9,*F, + ...=0,
after \ descriptions of the loop; and this would hold for all
integer values of \. As before, taking a sufficient number of
these equations for successive values of \, we infer that
F, = 0, Tr.= o, ... ;
if these are not evanescent, they would imply relations among the
members of a group, and so they can be satisfied only if
c,,= 0=...=c„ 0^ = 0=. ..c,, ....
Remove therefore the corresponding terms from the relation
CiW,+ ... + CmWm=0,
and proceed as before : we ultimately obtain zero as the only
possible value of each of the coefficients c. Hence no homogeneous
linear relation exists; the system is fundamental.
Some examples illustrating the preceding method of obtaining
the integrals of a linear equation will now be given.
Eai. 1. Consider the integrals of the equatioa*
in the vicinity of the origin. To obtain a regular integral, we take
* Tho equation is not in the exact form indicated in tlie tast. We have m  3,
ji, {0) = 0, and so a factor x haa been removed; alao ive have multiplied hy the factor
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98 EXAMPLES OF [40.
substituting, we hare
provided
= c,(a2l).o(«+l),
0=3Caa{a + 2)2Ci(a + 2)Co(a2)3,
the last holding for n. = 2, 3, . . ..
The indicial equation is
.(a!) = 0,
giving (simple) roots n = 3, a = 0, so that a f;ictor o + l can be neglected : the
relations among the coefficients are equivalent to
(a + 2nl)ca„^jC3„=0,
(a + 2«.)Cs,,5^„^.= "^;n (+3i^(f^2« + 2j'
Firstly, consider tlic root u = 2. We have
so that the int^ral belonging to the index 2 is
say the integral is u, where
Secondly, consider the root o = 0. From the original form of the relations,
by the first relation ; and the second relation is then identically satisfied,
leaving c^ arbitrary. Using the reduced form of the relations for the higher
coefficients, we have
and therefore the integral belonging to the index is
On subtracting c^ii from this integral, the reruainder is still an integral, and it
belongs to the index in the form (say)
Thus the system of two integrals, regular in the vicinity of x = 0, is
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40.] THE GENERAL THEORY 99
This method of dealing with the root a=0 is not quite in accord witii the
course of the general theory ; it happens to be successful because Cj is left
arbitrajy. In order to follow the general theory, we note that the coefficient
of Cj in the original differenceequation cootaina a factor a which vanishes for
the present root. Hence, taking
where A ia finite, and so on ; thus
»«(i+.!i)+'.'"{>+„f,+(.+Tf;s^j+}+.«('..).
where Ji(i>:, a) ia a holomorphic function of 3: which, by the general theory,
is finite when a = 0. According to the general theory, this quantity should
give rise to two integrals, viz.
Taking account of the value of c^, the first of thorn is zero, thus giving an
evanescent integral. The second is
or adding to this itit^ral ^Ca, which is an integral, we have
C{lx),
thus giving 1  a; as the integral.
Sx. 3. Discuss iT! a similar manner the regular integrals of the equation
a the vicinity of the origin : likewise those of the equation
x(lx)i'/'{l + i3!+%3^)«^ + {'3 + Ss:3^)w^0
a the same vicinity.
Ex. 3. Consider the integrals of the equation
n the vicinity of the origin. Subatituting the espresaion
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100 bessel's [40.
provided
for B = 1 , 2, 3, . . . ; these values give
where Tis a holomorphic fimotioii otx which is fiaite when a = l.
The indicial equation has a repeated root a = l; hence two regular
integrals are
The former is c^i^:, say the integral is u, where
the latter is c„a:logx+c„3^, say the integral is e, where
ii=a: log x + x^.
Both integrals belong to the index 1 ; and one of them must contain a
logarithm, since the index is a repeated root of the indicial equation.
Ei:. 4. Consider Bessel's equation for functions of order zero, viz.
Substituting
■u)=o^x''+c,x'^*'+,..+efie^*''+...,
we have
the latter holding for p=i, 2, 3, .... When those relatioi
value of iw is
— '• V (.+2)'^(.+2)(.+4)' ■■•;■
The indicial equation is a*=0, so that o=0 is a repeated root; thus the
integrals of the eqiuation, both of them helonging to the indes zero, are
M [S]„.
The firet. of them is
in effect, J^{x), on making Co=l. The second is
+«.fi2i^,(i+«+2rS7iii(i+H5)...).
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40.] EQUATIONS 101
Denoting this by Kf^ wheu Cu = 1 , we have
where ■^ {p) denotes the value of j {log n {:)} when s p. The two integralu,
regular in the vicinity of ^ = 0, are J^ and K^.
Ex. 5. Consider nest Bessel'a equation for functions of order b, via.
Dw=x^vf' + xv/ ■\{3^n?)vi=i!.
Substituting an expression
W = C„:C^iCi3!^*^+ ... +c„x'+"+ ...
in the equation, we have
provided
.,{(a + l}^«^} = 0,
for^=l, 9, 3, ... ; we thus have
r .0^ 3^ n
The roota of the indicial equation are
When n is not an integer, the oorresponding integrals are seen to he
effectively J„, J.,„.
When ji IB zero, we have a rei>eated root ; this case has heen discussed in
the preceding example (Ex. 4),
When n is an integer different from zero, the two roots belong to a group ;
and for a — — n, the coeffi.cient of afl^ is formally infinite, so that we have an
illustration of the general theory in §§ 36—38. We take the roots in order.
Firstly, let a= +«. : then the integral is
i (!)■
W n(p)n(«+ii")'
on taking e^ equal to — ^. This is the function usually denoted by J'„ ;
and it belongs to the index «,
Secondly, when a= n, one of the coefficients becomes formally infinite
through the occurrence of a denominator factor {a + ^nfit?. Accordingly,
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bessel's equation [iO.
and then
(„+a«)^_„3;^.ri
= C{(«+a«)^«2j^"ri^^— ,+ ... +{!)■
(a + 2)=ft=
n {(« + 2r)=^=}
say ; and now
Two integrals atiae through this root, viz.
Por the first of them, we have
SO that it provides no new integral. For the second of them, we have
raj,,] 1 2C, •■£Y''l' "'('''^y) ir
L8»J„— »'n(«i),;,W n(p) ■ "
say ; and
S?*''"<'°l="''['»(in) + 2l(„ + l)(» + 2)l.S]
+*^''A 'n(')'n'(wi '■■!'"■'+*'"+'■'» "■»©"
say ; so that the integral is M\+ W^,
In W^, the part represented hy
is a constant multiple of J„ and therefore can be omitted, owing to the earlier
retention of ./„. Eejecting this part, and taking
C=l2''"'n(j!,l),
the integral be(;omes
_/2Y%E_fcpzi)('"'Y'
U .;. n(j,) w
+ (f)"J. n4n'(i'+,) '"°^*'"H''»@"'
which differs, only by a constant multiple of ./„, from the espreswion given by
Haniel*.
* Math. AnH,..t. i (186fl, pp. 469—471, auotod iu my Treatise on Difurentinl
Equations, p. 167.
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40.] EXAMPLES 103
Ex, 6. Discuss in a similar manner the integrals of the equation
a(la)V + {l~(a + 6 + l)a'}w'(t6u^=0.
in the viciniti^ of ;k = 0, and a:=\ : indioating the form for the latter vicinity
when a+6="l.
This equation is the differential equation of the hypergeoinetric series
F{a, b, I, (b). When, in Logendre's equation
(l^)^3,^+y(y+l)».0,
the indei)endent variable is transformed to a:, where 2=1 — 2j;, it becomes
a:(la)«;" + (l2ar)V+i'(F + l)w=0,
which is the special case of the above given by 6=^+1, a= p. The
integrals of L^endre's equation in the vicinity of x=0 and of a;=l, that is,
in the vicinity of 5=1 and of b=1, can be deduced from those of the
hypergeometric equation ; the actual deduction is left aa an esercise.
Ex. 7. Apply the general theory to obtain the integrals of
j,^w"'  ^xhe" + Ixw'  8w = 0,
which are regular in the vicinity of x=0.
En. 8. Consider in the same way the equation
i)()«) = (l+^):cW'(2+4ic)icW + (4 + 10a!)«w'(4 + 12;»)«' = 0.
Substituting for w the expression
as in the earlier examples, we have
provided
C,>(« + al)(« + aS)^ + C„_l(« + a3)V'i+«4)=0,
for!i = l, 2, 3, ....
The roots of the indicia! equation are 2, repeated, and 1, so that they form
ft group the members of which differ by integers. Moreover, when = 1, the
coefficient Cj, which is
_ (a 3)^(a3)
'" aWW '
is formally infinite ; for that root, we shall take
o..C(.l)'.
Firstly, for the repeated root n = 2, we have
W^%X{\^{atfR{x,a%
where ii is a holomorphic function of x which remains finite when a = a . The
two integrals are
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101 EXAMPLES 0¥ [40.
it is easy to sec that they are constant multiples of
both of which belong to the itides 2,
Secondly, for the root a=l, we take a^=C(a—l)^, and theo
i)(w) = C(al)=(<.2)=:C",
""".
C{a
1)'
'^
G
1)'(.^2)'{.
.■( + 1)
— '.
— + (al)8(»,.),
where §(iK, a) is a holomorphic function of .r which remains finite when a = I.
In connection with this expression, three integrals are derivable, viz.
'«^' m.,, [SL.
The first of these is
C2x%
which is 2C% : it is not a new integral. The second is
which ia SCm^ ICu^: it is not a new iotegra,!. The third is
adding to it liOu^—ZiOui, the new expression is still an int^ral and is a
constant mulfcijile of %, where
which manifestly belongs to the iodes 1.
Ex. 9. Obtain the int^rala of the equations
(i) (l + a^3W(Z + i3:^)x^" + (i + lOx^)3^'~{i + 12^^)v> = 0;
(i!) (l + 'Lv}3^af'"(4: + 20x)xhci'" + {U + 72x)x^!i/'
which are regular in the vicinity of the origin.
Sx. 10. Consider tiie integrals of
Ihn=xw"'i.(aj+bjX+...)w" + {a^ + b^+...)ii/ + {aj + h^ + ...)ir/=0
in the vicinity of ^ = 0, the constant a, not being an integer. To obtain the
regular integrals, we substitute
provided
for values of n greater than zero, no one of the quantities /j,,/,, ... being of
degree in n greater than 2.
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40.] THE GENERAL THEORY 105
Tlie ioots of the indicial equation are
a=0, 1, 2a,.
Por a = 2a[, the difference equation determines coeificients c„, which
lead to a series converging for valuea cf ^[ within the common region of
convei^eni^ of the coefBcients of id", w\ i/>.
Por «=0 or 1, the differenceequation holds for values of n greater than
2 or 1 reapcctively ; the only other conditiona are
o 2 4 H 0
fo a= Hi
^ + =0
for a=l T) e tl e d fle'en e equat o again detem nes coefBcents wh ch
in each die leid to a Mr es thit on et^es w th n the lame regio a. the
aer e. that helongs to the esp nent 2 o Ea h of the latte tegnl b
a holo no ph funct on f and tl e efore the three te^iaU of the equat on
wh h are regula n the v c n ty of j. =0 are — one, a 1 olomo ph t n t on
of v belong ng to the dex i ^e ond i ke vise i holomorph f net on of %
belonging to the ndex 1 , and a th rd, belong ng t tl e ndex 2 ^
Ex. 11, Discuss the r^ular integrals of the equation in the preceding
example, when a, is an int^er.
Ex. 12. Prove that the equation
has m — i integrali \^hich are holoiiiorphie functinui of % in the vicinity of
t = 0, when Oj IS not an integer, the various coelhcients a,ib^a! + ,., in the
difterentiil e:iuation being holomorphic m that vicinity, and discuss the
regulai integrals when •/.^ is va integer (Poinoare.)
Ex. 13. Shew that the aeries
F( r^ = 1+^rl "(" + 1 ) _,.i.
ya,p,,r,r, ; ^^^^ ^2 ! p (p + 1) ^ {^ + l)r (r + 1) "^■■■
satisfies the equation
.(«,,„)J+a,;
and obtain the other integrals, regular in the vicinity of j; = 0.
Verify that, when a =r, the form of the function F, say (?()), tr,«), satisfies
the equation of the third order
and indicate the relation between the two differential equations.
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106 KEGULAR INTEGRALS [41.
Regular Integrals, fkee FJiOM Logarithms.
41. Alike in the general investigation and in the particular
examples, it has appeared that the regular integrals are sometimes
affected with logarithms, sometimes free from them. Thus if no
two of the roots of the indicial equation differ by a whole number,
each one of the integrals in the vicinity of the singularity is
certainly free from logarithms ; if a root of the indicial equation is
a repeated root of multiplicity n, then the first « — 1 powers of
log a; certainly appear in the group of n integrals which belong
to that root. When a root of the indicial equation, though not a
repeated root, belongs to a group the members of which differ
from one another by whole numbers, the integral belonging to
the root may or may not involve logarithms r we proceed to find
the conditions which will secure that every integral belonging to
that root is free from logarithms.
Let the group of roots be denoted hy p^^, p^, ..., p^,..., arranged
in descending order of real parts, so that p, — p„, for k = 0, 1, ...
fi — 1, is in each case a positive integer: and consider the root
p^. in order to obtain the conditions under which every integral
belonging to p^ shall be i'ree from logarithms. In the first place,
p^ must be a simple root of the indicial equation. Assuming this
to be the case, we know that the integral belonging to p^ is
in the notation of § 38. If we further admit the legitimate
possibility that, to this expression, we may add constant linear
multiples of the integrals which belong to the earlier roots
Poi Pi. i pi^i »nd still have an integral belonging to the root
p^, then, in order to secure that every integral belonging to p^
shall be free from logarithms, the . integrals belonging to the
earlier roots must also be free from logarithms; hence, as
further conditions, each of the roots p„, p^, ..., p^^, of the indicial
equation must be simple. These conditions also will be assumed
The full expression for the integral belonging to p^ is the
value, when a= p^,. of the expression
■[J;
KloS") 2 aTr£f'«' + +('»g'»)' ^ S.
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41.] FREE FBOM LOGARITHMS 107
in order to be free from logarithms, the quantities
for (7 = 0, 1, ..., /x 1, and for all values i' = 0, 1, ... ad inf., must
vanish : and if these conditions be satisfied, the above expression
will acquire the desired form. The conditions will be satisfied for
every value of u, if ^„(a) contains («— p^)" as a factor. But (p. 81)
g(«) ^(°) = rr /„^
3„(«) /o(« + l)/o(« + 2).../„(a + l') "'• *■
say; and ^o(a), which (§ 36) is equal to ?(«)/{«), contains
{a — py,)^ as a factor on account of its occurrence in f{'^)\
hence it is sufficient that H,{a) should remain finite (that is,
not become infinite) when o. = p^, for all values of v. Moreover,
ffo(o[)=l. Having regard to the equation by which ^^(a) is
determined, we obtain the relation
i/,/.(. + ^)+ir,_./,(. + — 1) + ...
All the quantities /^{a + v—l), ..., /„(a) are finite for values of a
that are considered; hence ff„/c(c[ + i') is finite if fl'„(=l), H^, ...,
/ff_i are finite, and therefore, on the same hypothesis, H^ will he
finite for all values of v, if it remains finite for those values of the
positive integer v, which make p^+v& root of the indicial equa
tion /5(^)=0. These values are known; in ascending order of
magnitude, they are
Consider them in ascending order. We have
R,.
When V =/5^_i — p„, a single factor
/.(« + .)
in the denominator vanishes when a^p^,,; and it vanishes to the
first order, because p„_i is a simple root of the indicial equation.
Hence, in order that jff„ may be finite ibr this value of v when
a=: pn, it is necessary that
hy{pi,) = 0, when v = p^^i — p^;
and it is sufficient that A„(p„) should vanish to the first order.
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108 REGULAR INTEGRALS [41.
When V = p^^  p^, two factors
in the denominator vanish when a = p^\ and each of them vanishes
to the first order, because p,_i and p^_j are simple roots of the
indicial equation. Hence, in order that H, may be finite for this
value of V when a = p^, it is necessary and sufficient that K{a}
should vanish to the second order when a = p„i the analytical
conditions are that
when V — pi.2~ Pii ^'Wd a — p^.
When p = p^_3 — p^, then the three factors
in the denominator vanish when a — p„; and each of them vanishes
to the first order, because p^_,, /Jbs, pnj are simple roots of the
indicial equation. Hence, in order that H^ may be finite for this
value of p when a = p^, it is necessary and sufficient that hy(a)
should vanish to the third order when a. = p^; the analytical
conditions are that
^ ' da da^
when v = pn^, — p^ anil a,~p^.
Proceeding in this way. we obtain the conditions for the
successive values of :' that need to be considered : the last set
is that
~/^ = 0, (<r = 0,l, ...,pl),
when V = pa~ Pk ^tid a = p^.
Such is the aggregate of conditions for a = p^. We have seen
that, in order to secure the freedom from logarithms of every
integral belonging to p^,, every preceding integral in the set as
arranged must similarly be free : and so wo have, in addition, all
the similar conditions ioT p^_^,p„^^,...,pi, there being no condition
for the simple root p^. When all these conditions are satisfied,
every integral belonging to p^ is free from logarithms.
Manifestly these conditions also secure that every integial
belonging to the roots /?„_,, p^i, ■■■, pi of the indicial equation
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41.] FREE FROM LOGARITHMS 109
is free from logarithms : (one integral, belonging to p„, is always
unconditionally free from logarithms) : it being assumed that each
of the roots />o, p], ..., ^^ is a simple root of the indicial equation.
The conditions thus secure that, when each of the /i + 1 greatest
roots in the group of roots of the indicial equation is simple, the
fj,+ l integrals belonging to those roots respectively are free from
logarithms.
The preceding investigation is based upon the results obtained by
Frobcnius, Cretle, t. LXxvl (1873), pp. 224—226.
A different investigation is given by Fucha, Crelle, t. lsyiii (1868),
pp. 361—367, 373—378 ; see also Tannery, Ann. de I'tc. Norm., t. iv (187(5),
pp. 167—170.
Ex. 1. A simple illustration arises in Ex. 1, § 40, for tlie equation
3!{2a?)w"  {x'+Ax.\.'i,){{\  x)w' i'w)=0.
With the notation of the text, wa have
/.(.). (.2),
p,% cI, PiO:
oonsidcr A„ i
;a)fora=p, = 0, v =
Pa
Pi
= 2.
?.(«)^
^('■)^o('')
^/,('' + l)A('< + 2)'
g.i")
"a"^^^^"'"''
*=(")= ^^^2/1 ("+ ^>'« <"+ ^^
= (4a)(a + l)(a+2)a.
The (one) condition m the j re ent t ise w that
A (qWO
when 0=0 : which minife^tlj is latished
Ex. 2. IfthcrtotH f tho indicia! e juati m are different from one another,
then the integrals whirl beloi g to them certainly possess terms free from
logarithms. (Fuchs.)
Ux.Z. Lebpo, pi p le the roota of tie mdiciil equation which form
a group, the members difieimg by itc^ois and nc two being equal; and
assume them ranged in descending ordei of icil 1 xrts. Denote pop, by
s—l; and form the eqmtion satisfied l>
V.*,('.);
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110
then according (
tion in W has nc
integrals of the c
by logarithms.
EXISTBKCE OF AN INTEGRAL
3 the indicia! equation for the singularity a.
negative roiita or has n^ative roots which
riginal equation in w are free from li^arithn
[41.
= of the equa
re aifected
(Fucha.)
En. 4. Shew that the integrals of the equations
(ii)
(iii)
where q and 6 arc constants,
free from logarithms,
the integrals of the equation
□ the vicinity of the origin. [They a
Ki^h'^f.]
42. If, instead of requiring (as in § 41) that every integral
belonging to an exponent p^ shall be free from logarithms, when
p^ is one of a group of roots of the indicial equation of the type
indicated in § 36, we consider the possibility that there shall be
some one integral free from logarithms, belonging to the exponent
and belonging to no earlier exponent in the group as arranged, no
such large aggregate of conditions is needed as for the earlier
requirement. Thus it is no longer necessary to specify that
po, ..., /3,^i shall be simple roots of the indicial equation; nor is
it necessary to specify that, even if these roots are simple, the
integrals associated with them are of the required form. The
conditions that arise will be particularly associated with a. = p^;
but they will be affected by modifications arising out of the possible
multiplicity of />„, ..., p^^i as roots of the indicial equation.
The detailed results are complicated : a mode of obtaining
them will be sufficiently indicated by an investigation of the con
ditions needed to secure that some integral free from logarithms
exists belonging to pi and not to pn, with the notation of ^ 36—38,
Suppose that p^ is a root of the indicial equation of multiplicity i ;
and let yi, ...,yi denote the set of integrals associated with p^,
where the expression of y,+], for « = 0, 1, ..., i— 1, is given by
P^l.
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42] FBEE FROM LOGARITHMS 111
It' Pi is a root of the indicial equation of multiplicity j — i, only
the first of the set of associated j — i integrals can be free from
logarithms: even that this may be the case, conditions will be
required. Denoting that first integral by W, we have
Now W certainly belongs to the exponent p;. Its expression, in
general, involves logarithms ; but there is a possibility of obtaining
a modification of its expression, so as to free it from logarithms, if
we associate with W a linear combination of t/^, ..., yi with con
stant coefficients ; and the modified integral will still belong to pi
but not to /Jo. Accordingly, consider the combination
where the constant coefficients A are at our disposal ; this gives
~ xp" 's i 2 !^ t+, , '' ,, (log ^y ^'Jf; xA .
(=i„^oj>=(i( ^'pi{tp)r ^ dp,*" j
What we require are the conditions that may, if possible, secure
that no logarithms occur in this expression for U.
The least aggregate of conditions that will secure this result
is ; first,
for all values of v, which secures the disappearance of (log*')'i
for all values of m and n such that pi + n = p„ + m, as well as
^_Sp.
0,
for p = 0, l,...,p„~pi—l, these conditions securing the dis
appearance of (log ic)*~' ; next,
.■(.1) ?'.»._
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112 CONDITIONS THAT AN INTEGKAL BE [42.
for all values of m and n such that pi + n = pt, + m, as well as
for p = 0, 1, ,.., p„ — pi
appearance of (log ic)'"^
ii! dpf
— 1, these conditions securing the
dis
; next,
= Ai_,g„ip,) + {i
2) A,
3»..
"3S
.^
2
2), SS.
such that pi + n
P. + "
., aa well as
for all values of m and )
for p — 0, 1, . .  , Pa — Pi — 1 y and so on. This aggregate is both
necessary and sufficient.
Manifestly any attempt to reduce it to conditions independent
of the constants A would be exceedingly laborious, even if possible.
The difficulty arises in even greater measure when we deal with
the conditions that some integral belonging to p^, where fj, > i, and
to no earlier index, should exist free from logarithms.
43. If we assume zero values for all the constants A^, ..., Ai
in the preceding investigation, the surviving conditions are cer
tainly sufficient to secure the result that the integral exists, free
from logarithms and belonging to its proper exponent: but the
conditions cannot be declared necessary.
The aggregate of this set of sufficient conditions is, in the case
of pi, that the equation
shall hold for «■= 0, 1, ..., i— 1, and for all values of n. As in
§ 41, it can be proved that all these conditions will be satisfied if
the equation
has a simple root equal to pf. Assuming this to be the case, then
an integral exists in the form
■■i.[ij....
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43.] FREE FROM LOGARITHMS 113
which is free from logarithms and belongs to pi (but not to p^)
as its proper exponent. If pi is a multiple root of the indicial
equation, the remaining integrals belonging to pi as their proper
exponent are certainly affected with logarithms.
Conesponding conditions, that are sufficient (but are more
than can be declared necessary) to secure the existence of an
integral, free from logarithms and belonging to an exponent p^
(but to no earlier exponent in its group), can similarly be found;
they are inferred from the investigation in § 41. If the equation
when n = p^^i — pn, has a simple root equal to p^; if the same
equation, when n = p^_5 — p^, has a double root equal to p^ ; if the
same equation, when n = p,^^ — p^, has a triple root equal to p,. ;
and so on, up to the case of n = pc — p^, when the equation must
have a root equal to p^ of Kiultipiicity fi. : then an integral exists,
belonging to pp. as its proper exponent (and not to any of the
exponents po, p,, ..,, p^^, and free from logarithms. If p^ is a
multiple root of the indicial equation, the remaining integrals
belonging to p^, as their proper exponent are certainly affected
with logarithms.
On the preceding basis, the identification of the integrals,
belonging to the group of exponents, with the subgroups as
arranged by Hamburger (§§ 23, 24) can be effected. The aggre
gate of integrals in the group, which are free from logarithms and
belong to their proper exponents, not merely indicate the number
of subgroups in Hamburger's arrangement but constitute the
respective first members in the respective subgroups. The
general functional forms of the remaining integrals belonging to
any exponent are (save as to a power of a factor ^Tn) similar to
those which occur in Jiirgens' form of the integrals in a sub
group*.
44. Ill the practical determination of the integi'ala of specified
equations, it sometimes"!" '^ convenient to begin with that root
* In tliie connection, tha followii^ memoirs may be consnlted: Jiii^ena,
CrelU, t. [JLXX (1875), pp. 150— 16B; SeMesinger, Grelle, t. oxw (1895), pp. 159—
169, 309—311.
+ As to this process, see the remarlia by Cayley, Coll. Math. Papers, t. viii.
pp. 458—162.
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114 cayley's [44.
among the group of roots which lias the smallest real part,
instead of beginning with the root that has the largest real part,
as in § 36, When the process about to be discussed is effective,
it has the advantage of indicating at once the number of integrals
associated with the group which are free from logarithms ; but it
is not always effective for this purpose, and it does not determine
the integrals that are affected with logarithms.
The equations determining the successive coefficients g,, </j, ...
in the expression
in the method ot Frobenius are (§ 33)
= ^«/„(a + K) +?.,/,(« + « l)+...+S'».A(").
for m= 1, 2, .... Let a group of roots of the indicial equation
/.(«) = 0.
differing from one another by integers, be denoted by p^, p,, ..., rr,
where a is the root of the group with the smallest real part ; and
replace a by «■ in the foregoing typical equation for the y's. Then,
whenever o + » is equal to another root of the group, the equation
in its given form ceases to determine ^„, as a unique finite
quantity.
It may happen that the equation is satisfied identically; in
that case g^ is arbitrary, as well as g^. It may happen that the
equation appears to determine ^„ as an infinite quantity : in that
case, we modify ^o as in § 36, and Qn is determinate after the
modification.
As often as the former ease arises, we have a new arbitrary
coefficient; if k be the number of these coefficients loft arbi
trary, then K is the number of different integrals, associated with
the group of roots and free from logarithms. These integrals
themselves are the quantities multiplying the arbitrary coefficients
in the expression
Ex. 1. As an example in which the process, of dealing first with the root
of a group that has the smallest real part, is efifeotivc as indicating the
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44.] METHOD 115
number of integriils free frum Ic^arithms, consider the equation
The indicial equation is easily found to be
(p2)(p3)S(p5} = 0,
th t tt re t ly ill 1 nteRral 1: 1 n n t the exponent 5, free
fml tlm th m b ml ntegral bei "ing to the exponent 3,
nd th e w 11 "« ta ly be an t gial, b 1 Dgin t that exponent and
aff ted th 1 gar th and tl re may be n t gi 1 belonging to the
ex[K) ent f ee f oia 1 gantl n
i di 1 t k th alue p=2 ad ub tt t
= + + + 'f'+
th j^ t t th mm diate p pose we d i>t consulor powers
higher thau fi m v, because p=5 is the root of the mdicial equation with the
highest real pait The equationa for determming the succes'<ive toefficients
0=Cj.O+eo.O,
0Cj(2) + c,(2) + to(l),
0Cs.O + C2(2) + c,(3) + Co(l);
from nhiL,h we see that to, <■„ tj lemam aibitrary 111 the other cociB
tioita tu^e expressible in terms of them CunsequeotlT, the equation haa
three mtegiais free fiom lonaiithms helonginn tn 2, \ 'i as their respective
propel expunents
(The equation was conotructed so as to haii,
for a fundamental system ; the system is easily derived by writing
when the equation for y is
^{l + j)y"'aS(7 + 83)y" + j'(29436«)y's(74 + 9fe)y' + (90+120«)^ = 0,
which can easily be treated by the general method of Frobenius.)
Ex. 2. As an example in which the process is ineffective, consider the
Taking, as usual,
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116
EXAMPLE
we have
provided
(p+n
for values
«=:
I, 2, ....
[44.
If instead, of beginning with the root p=3 fts in tho general theory (§§ 35,
36), we try p = 2, the equation for the coefficients c gives
«(«l)«.(»2)>0.„
determiuing c, apparently as infinite. To modify this, we take
the equation for Cj then becomes
(pl)(p2)c, = (p3)a0'2)(?,
which is satisfied identically, when p = 2. Thus c, rornaina arbitrary ; but
Co=0, The integral which would be obtained is, in fact, that which belongs
to p = 3; and the process is ineffective. There happens to be no integral
belonging to p=2 (and not to p=3) free from logarithms.
The .Tctual solution ia easily obtained by the general method of Frobeniua.
"We have
ir= (7.P ((p  2) + ^''~_f : + (p  2)2 (p  3)= R [z, f,)} ,
where R{z, p) is a holomorphic function of 3 when p is either 2 or 3; and then
For p = 3, we have the integral
For p = 2, we have the two integrals
M)3=ryn =Gz^JrCifi\(igs^ZCz\
The integral belonging to the index 3 is
free from logarithms ; that which belongs to the index 2 is effectively
2^+2^ log i,
which is affected with a logarithm, so that the index 2 possesses no proper
int^raJ free from logarithms.
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DISCRIMINATION BETWEEN SINGULARITIES
Discrimination between Real Singularity and Apparent
Singularity.
45. The singularity, in the vicinity of which the integrals
have been considered, is a singularity of coefficients of the equa
tion
da'" z — a di™"^ (z — a)™
and the indicoa to which the mtegrals belong are the roots of the
indicial equation for s = a, which is
00;
p(pl)(p<n + l} + p(pl)...{p
m+2)P,((j)+...
.■•+P.W0.
In general, tlie integrals of the equation in the vicinity of a cease
to bo holomorphic functions of s — a; thus they may involve
fractional powers or negative powers of z —a, and they may
involve powers of log (s — a). When this is the case, a is called *
a real singularity. If, on the contrary, every integral of the
equation in the vicinity of a is a holomorphic function of s — a,
then a is called an apparent singularity of the differential equa
tion. The conditions that must be satisfied when a singularity of
the equation is only apparent, so that it is an ordinary point for
each of the integrals, may be obtained as follows.
Let Wi, Wa, ..., w^ denote a fundamental system of integrals
in the vicinity of the singularity a ; and suppose that each member
of the system is a holomorphic function of 2 — a in that vicinity,
so that the singularity a is only apparent. Let A denote the
determinant {§ 10) of this fundamental system, so that
_ 1 d^^^tVi d''^^Wi dwi ^__ j _
dz'"~^ ' dz™~^ > ■•■> ^^ >
j rf™~'w2 d^~^Ws dwi
I dz"^' ' ^2™^ ' "■' "dz '
dw„
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118 REAL AND APPARENT [45.
and let Ar denote the determinant which results from A when the
column "  j  ^_,' is replaced by —r^ , (for s=l, ..., m). Then
as every constituent in A, and A is a holomorphic function of
^ a in the vicinity of a, both A^ and A are holomorphic func
tions oi z — a in that vicinity; neither of them is infinite there.
But as in § 31, we have
mi' <> "'>■
and some one at least of the quantities P^ (a) is not zero ; hence,
for that value of r,
A,(o)
a(»)
is infinite, and therefore
A(<.) = 0,
or the determinant of a fundamental system vanishes at an
apparent singularity. Moreover, as in § 10, we have
\dA^_ Pi(^) ^ _ A(tt) dQ(z a)
A dz z — a z— a dz '
where ff (s — a) is a holomorphic function oi z — a; whence
where A\s & constant. Now A is not identically zero near a, for
the system of integrals is fundamental ; hence A is not zero. We
have seen that A (a) = 0, and A {z) is a holomorphic function of
s — a; hence P, (a) m,ust he a negative integer, numerically greater
than zero. This condition is required, in order to ensure that a
is a singularity of the equation.
As each of the integrals is a function, that is holomorphic in
the vicinity of a, it follows that the respective indices to which
they belong must be positive integers ; and therefore the roots of
the indicial equation
Mpl)...(pm+l) + p(pl)...(pm+2)P,(tt)+...
... pP„_i(a) + P„(a) =
must be positive integers. (When one of these is zero, then
P^ (tt) vanishes.) Moreover, no two of these roots may be equal ;
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45.] SINGULARITIES 119
for otherwise, the expressions for the integrals that belong to the
repeated root would certainly include logarithms, contrary to the
current hypothesis. Accordingly, let the roots be p„ p^, ■■■,pm,
a set of unequal positive integers which we shall assume to be
ranged in decreasing order of magnitude : they thus form a single
group the members of which differ from one another by integers.
The integral belonging to pi involves no logarithm. In order that
every integral belonging to p^ may involve no logarithm, one
condition must be satisfied : it is as set out in § 41. In order that
every integral belonging to ps may involve no logarithm, two
further conditions must be satisfied ; they are as set out in § 41.
And so on, for each of the roots in succession until the last;
in order that every integral belonging to p^^ may involve no
logarithms, m — 1 further conditions must be satisfied, being the
conditions set out in § 41.
The aggregate of these conditions, and the property that the
roots of the indicial equation are unequal positive integers, give
the requisite character to the integrals. The condition that P](a)
is a negative integer makes a a singularity of the differential
equation. When all the conditions are satisfied, the singularity is
apparent.
In all other cases, the singularity is real
Ex. 1. Consider whether it ia possible that ,r = sliouid be
apparent singularity of the equation
where k and X are constants.
The first condition, that Pi(w) should he equal to a negative ii
satisfied : in the present instance, it is 4. To discuss the integrals
only ;
and substitute : then
I)v! = Ca{ai)(al)x
.,(a + ™4)(a + «l)={X(« + «l) + ^}c„_i,
The indicia! equation, being (a4) (a 1) = 0, has all its roots
positive integers ; so that another of the conditions is satisfied,
roots form a group.
equal to
The two
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120 EXAMPLES [45.
The integral, which belongs to the (greater) root 4 as its index, is a holo
morpbic function oi s: ; it is easily proved to be a constant multiple of (say) u,
where
"^t*^ 1.4 ^+ 1.4 ■ 2.5 ^+ 1.4 ■ 2.5 ■ 3.6 ^^j
= a^(l +71^4723^ + 73^ + . ..),
for brevity.
As regards the other root given by a=l, we have to assign the conditions
that the integral whioh belongs to it contains no logarithms. In accordance
witli the results of § 41, wo seo that there will be a single condition ;
expr^sing it in the notation there used, we write
Po=^< Pi = l' '•=PoPi=3. f=li
and we have to find h^{a) for i=^3, n=p,'=I. Now (§ 38)
/o(a) = (o4)(al),
f^ Mffoi")
/,(a + l)/„(a + 2)/,[« + 3)'
so that
A,(fl) = {X(a + 2) + ,c}{\(a + l) + «}{Xa + «},
Tho sole condition is that
and therefore we must have
K=\, or 2X, or 3\.
If K has any one of these values, the origin is only an apparent singularity of
the equation.
If K= X, the independent integi'al belonging to the root 1 is
If K= 2X, the integral is
If « =  3X, the integral is
In all other cases, the origin is a real singularity of the difierential equation.
The result, as to the relations between X and /i, can be verified inde
pendently. As w and u are solutions of the differential equation, we have
9M = T
v/'mi"=(^ + \\{wv/mi'),
and therefore
where ^ is a constant. Hence
dw \uj H* (l + yj_^+yiX^ + ys^ + ...f'
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If every integral is to be holomorphic in the vicinity of the origin, it is easy
to see that, as M belongs to the indes4, the only condition necessary is that the
coefficient of  on the righthand side should be aero. Thus
wliich, on substitution for yj, y,, y^, and mnltiplication by — 36, givM
thus verifying the condition obtained by the general method.
In this example it appears that the integral, which belongs to the smaller
root of the indioial eq^uation, is, in each of the three possible instances, a
polynomial in ix. It must not be assumed that such a result always holds
when a singularity is only apparent ; this is not the case*.
j&. 2. Prove that the origin is aji apparent singularity for the equation
where X and /i are constants ; and shew that no integral, holomorphic in the
vicinity of the origin, can be a polynomial in ;c unless ^ is a positive integer
multiple of X.
&. 3. Prove that 3=0 and z= 1 are real singularities for tho equation
?(lj)w"+(l2s)w')i'=0;
and that 2 = 1, := 1, are real singularities for
when n is an integer.
Ex. 4. Shew that s=r» is a real singularity for every equation of the
where /i {^} denotes a rational function of s.
£:e. 5. Shew that, if £=<o be an apparent singularity for each integral of
the equation
where P and Q are holomorphic functions of £~' for lai^e values of «,
then, if
!Pfj = Xlnegative powers of 2,
«©
* See some remarks by Gayley, in the memoir quoted on p. 113, note.
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122 EXAMpr,Es [45.
X must he a positive integer equal to or greater than 2, and. ft must be a
positive integer which may be zero. Shew also that, if X = 2, then it must be
Are these conditions sufficient to secure that each integral of the equation
is a holomorphic function of 3"' for large values of \z]'i
Ex. 6. Verify that every integral of the equation
is holomorphic for lat^e values of \z\.
Note on § 34, p. So.
To establish the uniform convergence of a aeries ^.g^^" for values of n,
Osgood shews {I.e., p. 85) that it is sufficient to h^ve quantities jl/„, indepen
dent of a, such that
provided the series SJf„ converges.
Take a circle in the aplane large enough to enclose all the regions round
the roots of /(p) = given by n — p"<j' — k' ; and let this circle be of radius
jj, BO that r^ is a constant independent of a. With the notation of § 34, take
constants C^, for values of i ^ t, such that
f (fi + O
while G^^T^ = y^. Then, as
^{v, + v)>M{a^v\ (,■, + .)•",(,(.■. + .)< /„(« + . + l),
for all values of v. Now, as in § 34 for the ratios of the r's, we find
and therefore the series
converges, li! being leas than It. Accordingly, by taking
the uniform convei^enee of the series ^ff^^ is established.
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CHAPTER IV.
Equations having their Integrals regular in the Vicinity
OF every Singularity (including Infinity).
id. We have seen that, if a linear ditfereiitiai equation is to
have all its integrals regular in the vicinity ot any singularity a, it
is necessary and sufficient that the equation should be of the form
cfe™ J— (I rfa™"' {z — af dz™^ " {z — a)™
in the vicinity of that singularity, the quantities P,, P^, ..., P,„
being holomorphic functions of s  ra in a region round a that
encloses no other singularity of the equation. We can immediately
infer the general form of a homogeneous linear differential equa
tion which has all its integrals regular in the vicinity of every
singularity of the equation, including .2=00. As Fucha was the
first to give a full discussion* of this class of equations, it is
sometimes described by his name; the equations are saidf to be
of Fachsian type or 0/ Fuoksian class.
Let a,, «£, ..., ttp denote all the singularities of the differential
equation in the finite part of the ^plane, and write
then the conditions are satisfied for each of these singularities by
the equation
d"'w_ S Qi, d'"'~'w
dz™ g=x 1^' dz™""
* See his memoir, Crelle, I. lxvi (1806), pp. 139—154.
+ Care mast be eiercised in order to discrirainate betweea eq'iiitiuas of Fuclisian
type and Fuehsian equations. The latter arise in eonneetion WLtb automorpMc
funotions and differential equations having algebraic coefficients ; aee Chap. x.
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124 EQUATIONS OF [46.
provided the functions Q, are holomorphic functions of z every
where in the finite part of the plane. To secure that the integrals
possess the assigned characteristics for infinitely large values of z,
we note that
+ = ..i!Q,
where iJ is a polynomial in — and is unity when a = qd , and
therefore
= £P"ii"
©.©•
where /ii is of the same polynomial character as B., and is unity
when 2 = CO . Now suppose that, for very large values of z, the
determinant A (s) of a fundamental system helongs to the index o,
so that
A(.)=^.r(l),
where 2" is a regular function of  which does not vanish when
2 = CO . Then, with the notation of S 31, we have
A.(»)=^— r.g),
where T^ is of the same character as T, save that it may possibly
vanish when z= k: taking account of the latter, we have
A.W^~ir.'(i),
where e is an integer ^0. Thus
..4='—©.
where U is a. regular function of  which does not vanish when
3 = 00 ; and therefore
Q.P.r
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46.] FUCHSIAN TYPE 125
for very large values of z. But Q^ is a holomorphic function of z
near z = oo ; this property, imposed on the preceding expression,
shews* that Q^ is a polynomial in z, of degree not higher than
(pi)..
Moreover, it was proved in the last chapter that all the
integrals of the equation
are regular in the vicinity of a = a, when the quantities Pi, ..., Pm
are holomorphic functions of s in that vicinity. Applying this
proposition to each of the singularities (including co ) of the
equation
with the restriction upon Q,, ..., Q„ as polynomials in s of the
appropriate degrees, we infer that all its integrals are regular in
the vicinity of each of the singularities (including x ).
Combining the results, we have the theorem, due to Fuchsf: —
When the m integrals in the fundamental system of a linear
homogeneous equation of order m. liave a^, a^, ..., ap as the whole of
their possible singularities in the finite part of the zplane ; and
when, all the integrals are regular in the mdnity of each of these
singularities, as well as for infinitely large values of z; the equation
is of tfie form
d^_G^ d^'w G^,p^ d"^w G'™,p_i)
de™' ^ ds'"'^ yjr^ rfs"'~^ ' 'ir'"
where ^ denotes ti (z — a^,and, Oi^if,!,, for /i = l,2, ...,m,is a
polynomial in z of degree not higher than fi(p — 1).
Conversely, all the integrals of this differential equation are
everywhere regular, whatever be the polynomials G and i/r of proper
degree.
Accordingly, this is the most general form of linear equation of
order m, which is of Fuchsian type.
' This result may also be obtained by using the trausformation zx — l and
applying to the equation, traasformed by the relations in g 5, the proper conditions
for the immediate vicinity of a; = 0.
t Crdie, t. i.xvi[186B),p. 146.
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126 EXAMPLES OF EQUATIONS [46.
Ex, 1. Legendre's equation is
(l32)W2sju'+m(n+l)ii'=0,
say
Its form siatiafiea alt the ueoesaarj conditions ; hence its integrals are regular
in the vioinity of i=l, z= —1, and are regular also for inflnitely large values
of 2.
Similarly, the hypergoomctric equation, which is
.(l.).J + {,(. + S + l).)«'a/5».0,
has all its integrals regular in the vicinity of ^ = 0, ! = 1, and regular also for
infinitely large values of z.
Eessel's equation of order zero is
__1 ,_£
 J «* ^"'■'
its integrals are r^ular in the vicinity of s=0 ; but, on account of the order
of the numerator of the coefficient of w in its fractional form, they are not
regular for infinitely large values of j.
The same result as the last holds for
which is JJesael's equation of order «.
A form of Lanie's equation, which proves useful {see Chap, ix, §§ 148 —
151), is
where A and B are constants ; its integrals are regular in the vicinity of any
point in the finite part of the 3plane congruent with 3 = 0, and these are all
the singularities in the finite part of the plane ; but they are not regular
for infinitely large values of s.
Ex. 2. The sum of all the exponents associated with all the singularities
(including oo ) of the equation of Tuchsian type obtained at the end of the
preceding investigation is the integer (p l)ra{m — 1), a result first given by
Fuchs*. The proof is simple.
The polynomial ^pi is of order not higher than j) — 1 : say
G„i = A^^ + ....
The indicial equation for the singularity a„ is
ji'^)/]C/9_n...(Sm + 2) + ...,
' Crdle, t. Livi, p. 145.
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46.] OF FUCHSIAN TYPE 127
the unexpresBed terms on the righthand aide constituting a polynomial in 6
of order not higher than m — 2. Hence the sum of the indices for the singu
and therefore the sum of the indices for all the singularities Oj, a^, ..., (tp in
the finite part of the plane
*'(""+ iT?sr
.Spm(»1) + J,
because Oj, a.2, ••■, ^p are tha roots of i^ = 0.
The indices for co are obtainable by suiistituticg
the indicial equation for t» is
{■irpij>+l)...{p+ml)={l)''Ap{p+l)...{p+m2) + ...,
so that the sum of the indices for co is
im(ml}A.
The total suqi of all the indices is therefore
Hfl)M(M~i}.
Ex. 3. The general eqtiation of Fuchsian type, which haa all ite integrals
regular in the vicinity of every singularity (including to ), has been obtiiined.
The limitations upon tlie form of the type are mainly as to degree, so that
generally the construction of the equations, when definite singularities and
definite exponents at the singularities are assigned, will leave arbitrary
elements in the form. The instances when the equations are made com
pletely determinate by such an asaignment are easily found.
Taking the equation as of order m, we have polynomials
o,_,W, o,_,(.) ff„.(.)
which, in their most general form, contain
pf(2pl) + (3p2) + ...(mpmHl)
= ip™(™il)^m(™l)
The assignment of the positions of the singularities merely determines V' :
it gives no assistance to the determination of the constants in the poly
nomials ff.
Each of the /> singularities in the finite part of the plane requires ™
exponents, as does also the point £ = ai ; so that there are ™{pl) constants
thus provided. But, by the preceding example, their sum is definite : and
thus the total number of independent constants thus provided is
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128 EQUATIONS OF FUCHSIAN TVPE [46.
If therefore the equation is to bo made fully determinate bj the assign
ment of these constants, we must have
and therefore
ipm(™l)=J(ml)(M+3).
When m=l, p can have any value; that is, any homogeneous hnear
equation of the iirat order, which has ita integral r^ular in the vicinity of
each of its singularities and of s = oo , is completely determined by the aaaign
ment of singularities and of the exponents for the integral in the vicinity of
the singularities.
For such equations of the first order, let a,, ,.,, Op lie th ng 1 t a
the finite part of the plaoe ; let mj, ..., m^ be the ind to wh h th
integral beloi^s in their respective vicinities, and let be the d x f
2=00, so that m+ 2 ^^ = 0. The equation is
which gives the indes for : = so as equal to  2 m,. , beiiig it* proper value.
■yiTien tn>l, then
ao that, as p is an int^er, m must be 2 and then p=2. Thus the only homo
geneous linear equation of order higher than the first, which is of the
Fuchaian type, and is completely determined by the assignment of the singu
larities and of the exponents to which the integrals at the singularities
belong, is an equation of the second order : it has two singularities in the
finite part of the plane, and it has a = co for a singularity ; and the sum of the
sis indices to which the integrals belong, two at each of the singularities, is
^(21)2 (3  1), that is, the sum is unity.
The discussion of the determinate equation of the second order of the
foregoing type will be resumed later (§§ 47 — 60).
IiTole. If p =0, so that the equation has no singularities in the finite part
of the plane, the coefficients are constants if the equation is to be of Fuchsian
tyX>e. The only singularity of the integrab is at co .
If p=l, m>l, the number of arbitrary constants is less than the number
of constants, due from the assignment of the indices at the finite point and
at 3= CO : the latter cannot then all be assigned at will.
For values of p greater than 1 and for values of m greater than 1, the
number of arbitrary constants in a linear differential equation, which are
left undetermined by the assignment of the singularities and their indices, is
= ipm{m + l)im{«,l)^{m{p + l)l}
=i(™l){™(pl)2},
which for all the specified values of p and m, other than m = 2 and p = 2 taken
simultaneously, is greater than zero.
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Ex. 4. Consider the equation, indicated in the Note to Es, 3, all whose
integrals arc regular at the only finite singularity, which can be taken at the
origin, and regular also at infinity ; it is
'd^" z d^^ "^ ^ (fe™2 +■■+^"'11
where /j./j, ...,f^ are constants. The assignment of indices for s = determ
ines ^, ..., ^, and 80 determines the indices for 3 = co; and similarly the
aesignment of indices for z = 'a determines those for 2 = 0. In fact, the
indicial equation for ! = is
p{pl)...(pW + l) = J^p(pl).,.(p™ + «l)/„
and the indicial equation for : = (c is
( \TB{6\\) (fll r=2( l)'""fl{fl + l) {fi+ K + 1)/"
t t on e BY deot th fc the ts c n bo a a d 5 a r on fr ta 1
equat n m the form p+5=0
As re^rd the ntegrals, t ea y to vo fy n a cordance w th the
general theory that the ntegral wh h belongs to a s mple r ot of the
nd al ejuato for =0 a c n t t miltjlo of and that the
ntegraL wl h bel n t j t i le x)t of tl t e iiiat o re tant
mult pies of
for a = 0, 1, ...,«!.
Es;. 5. Consider tlie equation
i>ip = 3{l3)w" + (l2s)w'iw = 0,
wMch* clearly satisfies the conditions that its integrals should be regular,
both in the vicinity of its singularities and for large values of :.
To obtain the i
utegrals in the vicinity of s=0, we substitute
)w = eo2"fc,i!' + i + ... + c„s" + ''f...,
and find
zDiB = a„aH'',
provided
{a + «')^c„ = (o+«^)^c„i ;
so that, writing
f(. + ^){a + )...(a + mi)) =
the value of lu is
«> = c„s"(l+yi! + y22^ + ...).
* It ia the differential ec[uatioa of the q^uarterperiod in elliptie functions : fo
detailed discussion of the equation, See Tannery, Ann. Ae Vka, Noi'm. Sup., S^r. S
t. viii (1879), pp. 169—19*, and Fuohs, Crelle, t. lxsi (X870), pp. 121—136.
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130 EQUATION OF QUARTERPERIOD [46.
Tlie indieial equation is o*=0 : accordingly, the two integrals belong to the
indes 0, and they are given by
[a..
To particularise the integrals, we take c^^^^tt ; the first of the integrals then
becomes
.<„=i.{,.(iy..(H)'..)
say : and the second of them becomes Z (s), where
 1 2 *" 2toJ
say, where
P™ 1 2^3 4'^"'"^2»ii 2m'
And now the two integrals in the vicinity of the origin are
K(.}, L{f,).
To obtain the integrals in the vicinity of ^=1, we substitute
when the equation takes the form
which is of the same form as in the vicinity of ^ = 0. Accordingly, the
integrals in the vicinity of 2=1 are given by
TW, !■{')■
To obtain tbe integrala in the vicinity of 3 = aj, wo substitute
1
when the equation takes the form
The indicial equation for (=0 is
«("i)+iOi
and we find the equation for « to bo
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46.] IN ELLIPTIC rUNCTIONS 131
of the same form as in the tirat aad the second eases. Accordingly, tha
integrals of the original equation in the vicinity of a — oo are
The integrals are thus regular in the vi n ty of tl e t! ve nj, 1 r t cis
0, 1, to. Of these, the integrals K{z) i(, ) are s gmfl ant n the ioma n
e<I, aay in D^; the integrals K{a!) L{x) ore s gi fic^nt n the domain.
a; = zl<I, say in iJi ; and the integiaU i*A if) ^L{t) aie 3 gn ficant in
the domain \t\ < I, that is, \z\ > 1, aay in 0„. The ser es A ( ) d verges wi en
2= I, so that the integrals cease fco be significant for such a value.
The domains D„ and Di have a common portion, so that values of z esist
which are defined by
a <1, 1J1<L
Within this common portion, the integrals K{z), L{z), K{a:\ L{x) are
significant : so that, as K {£) aud L (s) make up a fundamental system, we
K{^)=AK{z)^BL{z), L{:>^)=A'K{z)+B'L{^),
where A, B, A', B' are constants. The values of the constants are determined
as follows by Tannery.
The integrals are compared for real values of a which are positive and
slightly less than 1, so that, as s then approaches 1, K{e) tends to an infinite
value. To obtain this infinite value, we note that, as
by Wallis's theorem, wo have
and therefore, for I'eal values of s between and I, we have
The difference of the two quantities, between which the value of K{z) lies, is
which increases as the real value of s increases and, for a = l, is
that is, 1  log 2, Kenofi wc may take
.,(.)5iog(i.).
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132 EQUATIOK OF QUARTER PERIOD [46.
where
^^>.(3)>j7rl + log2;
find the values of z are real, positive, and less than 1. The result shows that
K(s) is logarithmically infinite for s=l.
Proceeding similarly with I{z) in the expression for L (s), we have, for real
values of z between and 1,
The difference of the two quaatitics, between which the value of iHa)
s the real value of 3 increases and, for 3=1, is
and therefore the foregoing difference is less than
that is, less than (1  log 2) log 2. Hence we may take
where, for real positive values of z that are less than 1,
0<('W<(llog2)log2.
The expression can be further modified. Wo have
S ^^"'<log3 I ^,
for the values of z considered. The difference between these two series is
I log 2^^
the real value of s increases and, for 3=1, is
S ^ (log 2 £{„).
8 =1 '+.
"^ 2m + l 2111 + 2
1
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«.]
IN ELLIPTIC FUNCTIONS
aod therefore the difference is
<.!,i^i)<^(l>'>l!^)'
oil evaluating the ai
sries. We may therefore take
J^5^=J_^'^log2("W
where
= log(l^)log3^'(^),
0<t"(s)<aalog2).
Therefore, finally, w
'0 have
where
so that
i/(.}=ilog(l^)log2.,(4
0<<i(s)<l(log3)^;
and the values of s
considered are real, positive, and less tl
In the region co
mmon to i>„ and jD,, we have
KIf)AKif)+BL(i).
and therefore, for real values of z less than (but nearly equal to) 1, that is, for
real, positive, smaU values of x,
^(3^)^^*(a)i^log^2fllog^log24Cf,W+5{.(s)ilog;i^}loge.
When s tenda to the value 1, the term log^c log 2 tends to the value : more
over, K{!0) then tends to the value ^w ; hence, taking account of the infinite
terms on the righthand side, we have
^145 log 2 = 0.
Again, when % is real, small, and positive, m ia real, positive, and loss than
(but nearly equal to) 1 ; hence
4W..(»)^ilog(l»).,(l.)ilogz,
all the terms in which
when l^l is small.
{i^)iiog2=.iArw+sir(3)ioi
finite except those i
J+S/(s>,
volving lo
a holomorphio function of s; thus
A =1 log 2 ;
o that A and B a
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134 EQUATION OF QUARTERPERIOD [46.
Similarly, for the other equation
for values of x and s iu the ooramon region, wo have, for real, positive values
of z less than I, that is, for rea], positive values of x that are small,
^Wlog»+/W^'{,(.)iIog(l.))4ir910E(l,)log2 + ,,(.)l,
hence, taking account of the logarithmically infinite terms on both sides, we
^' + 4S'log2 = ff,
Nest, tating the same equation for values of z that are small, real, and
positive, 80 that x is real, positive, and less than 1, we have
xir(.)+ff{iwios.+/(,)).;rWiog «:+/«.
When a; is nearly unity,
jrM.,Miiog(i»),
SO that K{3:)logx, for a; nearly equal to 1, is small : and it vanishes when
a'isl. Also, for those values of ^,
= 21og3log24.,(^);
whence, equating coefficients, we have
ijrff=21og2.
Thus
5'= log 2, ^■=(log2)=w.
Accordingly, when j lies within the portion common to the two domains Df^
and Z)j, defined by the relations
irw.(iiog3)^(.)ii(.) I
iM.{"(iog2)>,}irM(i log 2) L(.) \
where x=l~e.
These results shew that, for complex values of j such that [zj^l, both
X(i) and Z(2) converge. The fiist of them is a known result in the theory
of elliptic integrals ; writing b = *^ ^^^ Z(j)=£, AT (^) = Jf', we have
an equation which is specially useful for small values of h Similarly, for
Ta]u©a of i nearly equal to unity, we have
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46.J RIEMANN S P FUNCTION l3o
Ex. 6. With the notation of the preceding esample shew that, for values
of z common to the domains /), and />„ as defined by
the integrals Z'{a^), i (a'), t^KU), i*Z(;) are connected by the relations
(Tannery.)
E^. 7. Denoting the integrals of the equation in Es. S that are associated
with the values 2 = 0, 1, cc. by ^, Z ; K', L' ; K", L" ; respectively ; denoting
also the efiect upon a function Uof a, simple cycle round a point ahy \_U^,
and of simple cycles round a and b in succession by [ U^r, , prove that
[^li^fs+f logaW'+^'i';
and express [L%, [L"],,, in terms of K', JJ. (Tannery.)
Ex. 8. Discuss, in the same manner as in Ex, 5, the integrals of the
(i) ^(l~.)W'i^=0;
(ii) 2(lz)«7" + (lj)«/ + J«.=0;
(iii) 2(l^)W' + w'iw=0.
RiEMANN'S PFUNCl'ION.
47, It has already been proved {Ex. 3, § 46) that the only
linear differential equation of any order other than the first, which
is made completely determinate by the assignment of ita singu
larities and of the exponents to which the integrals belong in the
respective vicinities of those singularities, is an equation of the
second order which, if ifc have oo for a singularity, has two other
singularities in the finite part of the plane. If the latter be
at h,, k, then the transformation
z—h_ho—b x~a
z~k k c—a x~b
gives a, 6, c in the aiplane as the representatives of h, k; oo in the
zplane. The transformation manifestly does not aifect the order
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136 RIEMANN'S [47.
of the equation, its sole result being to make a, b, c (but not now
00 ) singularities ; we shall therefore suppose this transformation
made. Accordingly, we proceed to consider the properties of the
function, which thus determines a differential equation ; they
depend upon the properties initially assigned, which are taken as
follows.
In the vicinity of all values of s, except a, b, c (and not
excepting od when a, b, c are finite), the function is a holooiorphic
function of the variable.
In the vicinity of any point (including the three points a, b, c),
there are two distinct branches of the function ; and all branches
of the function in the vicinity of any point are such that, between
any three of them, a linear relation
AT + A"P" + A'"P"' =
exists, having constant coefficients A', A", A'". (So far as this
condition affects the differential equation, it manifestly determines
the order as equal to two.)
As exponents are assigned to the three points, let them be a
and a' for a : /3 and (3' for 6 : y and 7' for c ; these quantities
being subject (§ 46, Ex. 2) 60 the condition
a + a' + ^ + S' +y + y=l.
It further is assumed that a — a.', ^ ~ ^, y — y ai'e not equal to
integers. The branches distinct from one another in the respective
vicinities are denoted by P„ and P„. ; Pg and Pp ; P^ and Pyf.
From the definition of the exponents to which they belong, the
functions (s — a)'P^ and (s  ra)"'P„' are holomorphic in the
domain of a and do not vanish when z — a. Similarly for b and c.
After the earlier assumption, it follows that any branch
existing in the vicinity of a can be expressed in a form
where c„ and c^' are constants ; and likewise for branches in the
vicinity of b and c. The assumption made as to a— a.', /3~0',
7 — 7' not being integers will, by the results obtained in §§ 35 — 38,
secure the absence of logarithms from the integrals of the
differential equation: it manifestly excludes the possibility of
either of the branches P^ and P^', Pg and P^', Py and Py, being
absorbed into the other.
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47.] pFUNCTioN 137
Biemann* denotes the function, which is thus defined, by
y
and the function itself is usually called Etemann's Pf unction. It
is clear that a and a.' are interchangeable without affecting P;
likewise yS and jS' ; likewise j and 7'. Also, the three vertical
columns in the symbol can be interchanged among one another
without affecting P ; six such interchanges are possible. Again,
if P be multiplied! by (a:  afixb)^' {x c)', the effect
is to give a new function, having a singularity at a with expo
nents a H S, a' + 8 : a singularity at b with exponents — B — e,
iS' — 8 — e ; and a singularity at c with exponents 7 + e, 7' + e.
Every other point (including 00 ) is of the same character as
for P. Hence
/ a, w r<* ^ '^
ia:hr
7
fS /3'
e 7+6 Jr,
ey + e J
the exponents on the righthand side still satisfying the condition
that the sum of the exponents shall be equal to unity,
A homographie transformation of the independent variable
can always be chosen so as to give any three assigned points
«', b', c' as the representatives of a, b, c. Accordingly, let such a
transformation be adopted as will make a and 0, b and <x> , c and 1,
respectively correspond to one another : it manifestly is
The indices are transferred to the critical points 0, 00 , 1 ; every
other point is ordinary for the new function, as every other point
was for the old. For brevity, the transformed function is denoted by
' Ges. IVerke, p. 63.
t The sum o£ the indices in the factor
a singularity for the new function.
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where the twoterm columns are to be associated with
order. Also, sir
ice
— 5tH.
it follows that, i
ixcept as to a constant factor.
(»»)'<«c)
(«!,)•+. '""' " *' ">
agree ; and thus
i, as regards general character, we have
a!''{l~wyF
1" 1^ 7 A pC +S /5 «« T +
W^'7 / U'+S ,S'S6 7' +
As a — a', /3 — /3', 7 — 7' are the same for the Pfunction on the
righthand side as for the Pfunetion on the left, Riemann denotes
all functions of the type represented by the expression on the
left by
P (««', 0^', 77', ^').
In the transformation of the variable, the points a, b, c were
made to be congruent with 0, » , 1 in the assigned order. A similar
result would follow if they had been made congruent with 0, 00 , 1
in any order or, in other words, if 0, 00 , 1 be interchanged among
themselves by horaographic substitution. As is known, six such
substitutions a
or, taking account of the association of the exponents with the
first arrangement, the table of singularities, exponents, and
variables for the six cases is
Oool Owl Oool
a 13 y x'; 7 /3 a 1^'; /^ « 7 ^.i
a' ^' i i ^' d ^' a' y'
7='^i'; «7/3 ^3^ ; ^ "y " 1 _"^ '
y' a ^' a! 7' /3' /3' 7' a
so that Pfunctions of these arguments with properly permuted
exponents can be associated with one another.
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48,] i'FUNCTIONS 139
48. The significance of the relation
a + <x' + ^ + 0' + y + y' = l,
in connection with the function, appears from the following con
siderations. When the singularities are taken at 0, x , 1, the axis
of real variables, stretching from — co to + co , divides the plane
into two parts in each of which every branch of the function is
uniform; or, if the singularities be taken at a, b, c, then a circle
through a, h, c divides the plane in the same way. In either ease,
taking (say) the positive side of the axis or the inside of the
circle, the linear relations among the branches of the function
give
P, =B,Fe + B,P^.] P. =(7.P^ + 0,Py)
P„ = B,'P^ + B,Te. j ■ P,< = C'P^ + a^I\. J '
Bay
P„, P„ = iB,,B, $Pp, P^.) = (i^P?, Ps'),
I Pi', p; I
p., p. = ( 6', , C, 5P„ Py) = {cfP„ Py) ;
I o;, a; \
and with the usual notation of substitutions, lot
Pg,P^, = (65P„, P,,),
.Py,Py^ = (ciP.,P.,).
Consider the effect upon any two branches, say P„ and P,', of
circuits of the variable round the singularities.
When it describes positively a circuit round a alone, they
become e^'''°P^ and ^'^' P^i respectively, so that, in the above
notation,
P„ P,; become ( e^^, $P„, P.).
I 0,«''
When it describes positively a circuit round h alone, then P^ and
P^i become e'"''^Pp and e^'^'P^^ respectively ; and therefore
P„ P„. become (fije^'^, J^jPa, P.')
I , e^'l
Similarly, when it desciibes positively a circuit round c alone,
P., P„ become (cfe'^y, fc'^^P., P.').
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140 RIEMANN'S i'FUNCTION [48.
Accordingly, when z describes a simple circuit round a, b, c, tho
initial branches P^, Pa' are transfonnod into branches
(oj.., VcmeO. jsf,,, JP..P..),
, 6''"T'[ I , (f"'f' I , e=°'"'
s.y nP.. P..).
Such a circuit encloses all the singularities of the functions ; and
therefore* each of the functions returns to its initial value at the
end of the circuit, so that
(7)=(1, 0).
o. l
The determinant of the righthand side is unity; hence the
determinant of / is unity, and it is the product of the determinants
of all the component substitutions. Now as (c) and (c)"' are
inverse, the product of their determinants is unity ; and likewise,
the product of the determinants of (b) and {b)~^ is unity. Hence
we must have
an equation which is satisfied in virtue of the relation
a+a' ^ + (i' + y + y'^l:
the sum of the exponents could be equal to any integer merely so
far as the preceding considerations are concerned.
In the present instance, the property, that a function returns
to its initial value after the description of a circuit enclosing all
its singularities, can be used in the form that the effect of a
positive circuit round c is the same as the effect of a negative
circuit round a and round b. Applying this to P^, we have
C,Py^^ + (?,Pye^'" = e="" (^i^pe^" + BJ^^.e"^'^) ;
and i'rom the expressions for P„, we have
C\P, + C,Py = B,Pff + B,Pp..
As Pp and P^' are linearly independent of one another, it follows
that e*i™*— eV"* must not be zero, that is, 7 — 7' must not be an
integer. Similarly for a — a' and 0 &■
Ex. Prove, by means of those relations, that
O,'" S/8in(a'(^+y')ir Bj'sin(a'+^'+y),r '
£= >')*_ A«i°('' + g + y)T _. ga'^iti(n + g' + v)^
G4 £,'8m(a' + jy + v)n iJ2'sin(a'4(3' + r)7r"
(Eiemann.)
* T. F., % yo.
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49.] determines a differential equation 141
Differential Equation determined by Eiemann's
pfunction.
49. As regards the differential equation, associated with these
Pfunctions, and determined by the assignment of the three
singularities a, b, c, and their exponents, we know that it must be
of the form
d\v A'z' + B'z + C
d^'^{,~a){zb){zc)d^^ {zaf{zbf(zcr
which (§ 46) secures that a, b, c, x are points in whose vicinity
the integrals are regular. Now the singularities are to be merely
the three points a, b, c, so that oo must be an ordinary point of
the integral. Taking the most general case, when the value of
every integral is not necessarily zero for s = x , we have an
integral
where Kt, does not vanish. Substituting, we havu
the unexpressed terms being lower powers of 2; hence
(2  A) K, + A"K, + \B" + 2A" (a + 6 + c)] K„ = 0,
and so on. Using the result that A" = 0, the equation may 1
written in the form
d'w f A B G\ dw
U^'^\Ta^~zb^zc}~dz
Forming the indicial equations for the singularities, we have
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142 DIFFERENTIAL RQUATION DETERMINED [49.
as the indicial equation for a ; and therefore, as its roots are to be
a and a', it follows that
A = 1 —a— a', X= aa' {a—b)(a — c).
Similarly
5 = 1/3/3', fi = 00'iba)(bc).
C =1 —y— y, V — 77' (c —a)(c — b).
Moreover
A'=A+B + G=%
on account of the value of the sum of the six exponents; tlio
condition
is thus satisfied by B"= 0. All the quantities are thus determined,
and the equation has the form*
rf'w naa' lfi^'ljy'\dw
dz^ \ za z—h zc ) dz
[ <.a'{ah){ac) &0 (ba){b o) 77' (c  a) {c  6)1
from the mode of construction we know that the integrals are
regular in the vicinity of the singularities a, b, c, and are holo
morphic for large values of s. This is the differential equation,
vith (and determined by) the function
(a b c
W /3' 1
The branches of the integral in the vicinity of a are P, , P„;
those in the vicinity of b are Pf,, P^i ; and those in the vicinity of
c are P,, P,.
Passing to the ibrm of the function represented by
U' /3' 7' '
where the three singularities are 0, =0 , 1, we deduce the associated
differential equation from the preceding case by taking
c( = 0, &=«, c = l;
• Firat given by Papperitz, Math. Ann., t. xiv (1865), p. 213.
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id.] BY RIEMAKN'g PPUKCTION 143
after a slight reduction, the equation is found to be
d'w liia'(l+i3 + /3');; <i»
((«■"'' 2(12) i
, W(aa' + gff'T7')2 + gi3V „
+ 2(l2)
The branches of the integral in the vicinity of the origin are P,,
P^', so that ^""'Po, z~°''P^' are holomorphic functions of s, not
vanishing when 2 = 0; those in the vicinity of a = 1 are Py, Py,
so that {z — l)'"'Py, {s — l)~*'Py' are holomorphic functions of
z—l, not vanishing when 2 = 1; and those in the vicinity of
s — <rj are Pg, Pp., so that z^P^, s^'Ppare holomorphic functions of
— , not vanishing when z=<r^ .
Lastly, passing to the form of the functions included in
P(t.«'. /S/?, 77. »).
we saw that they arise from the association of arbitrary powers of
s and 1 — 2 with the above function in the form
and that they lead to a function
/. +S, ftS,, y+. '.
^W + S, ^'B,, 7' + .'/
Thus we can make any (the same) change on a and a' and, as they
are interchangeable, we can select either for the determinate
change; accordingly, we take
say, as the modified exponents. Similarly, we can make any (the
same) change on 7 and j : we take
ry —y = 0, J — y = V — ^ — /i,
say. Then the new values of the exponents for co are
/3 + a + 7, \ say,
and
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144 INTEGRALS OF THE EQUATION OF [49.
on reduction : or the exponents are
0, 1  /' , for 2 = ,
\, /t , for s = CO ,
0, v~\fi, for s = 1 .
Their sum clearly is unity : moreover, with the preceding hypo
theses, the quantities 1c, /^ —X, v \ — fi are not integers.
Specialising the last form of the equation by substituting this set
of values for a, a', &, ^', 7, 7', we find the equation, after reduction,
to he
which is the differential equation of Gauss's hypergeometric series
with elements X, /t, p. Either from the original form of the
Pfunction, or from the resulting form of the equation, the
quantities X and /j. are interchangeable.
50. Taking the equation in the more familiar notation
, dhu , , ^ , , 1 dw , „
so that tho exponents are 0, 1— 7, for z^O; a, 0, for 3=x;
0, 7 ~ a — (8, for ^ = 1, we use the preceding method to deduce the
wellknown set of 24 integrals.
Denoting as usual by F{a:, 13, 7, z) the integral which belongs
to the exponent zero for the vicinity of z = 0, we have
«{« + l)M8 + l)
1.2.t(t+1)
assignitig to the integral the value um'ty when z= 0. If
z'(i^yF{cL',^;y',z)
be also an integral, then the exponents for each of the critical
points must be the same as above ; hence
S, B + ly =0, 17 , for 3 = ,
e, e + j'a'^'^O, 7a/3, for 2 = 1 ,
a'Se, (3'Se =«, /3 , for s = oo.
Apparently there are eight solutions of these equations ; but as a
and y3 can be interchanged, and likewise a' and 0', there are only
four independent solutions. These are : —
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50.] THE HYPERGEOMETEIC SERIES 145
I. S = 0, e = ; giving a' = a, ^' = /3, 7' = 7 ; and the
integi'al is
II. S = l7, e = 0; giving a' = l + «7, ^' = 1+^^,
7' = 2 — 7 ; and the integral is
2'rf (Ha7, 1+^7, 27, ^);
III. S = 0, e = 7 — a — /3; giving a' = 7 — n, /3' = 7 — /3, 7' = 7 ;
and the integral is
(l_^)v— fljr(^_a, 7A 7, ^);
IV. S=l7, e = 7a^; giving a' = 1  /3, ^' = 1  a,
7' = 2 — 7 ; on interchanging the first two elements,
the integral is
^'T (1  z)yf F(l  «, 1  A 2 ~ 7, z).
Next, it has been seen (§ 47) that, in the most general case,
Pfunctions can be associated with a given Pfunction, when the
argument of the latter is submitted to any of the six homographic
substitutions which interchange 0, 1, x amoDg one another,
provided there is the corresponding interchange of exponents.
Taking the substitution e'z = 1, the new arrangement of exponents
a, , for / = 0,
0, 7a/?, for s' = l,
0, 17 , for /=co;
heirce, if
2"(l«')f(«', /3',7'»)
is an integral, we must have
S, S + 1  7'  0, ,3
for /O,
e, e+7ci'/3'0, iafi,
for s'l,
a8e, /3'S. 0, 17 ,
fop /«.
Again tlrere are four independent solutions ; they are i —
IX. Sa, «(); givingf'a,/3'l+«
and the integral is
 7, 7'  1 + « 
13
zfU 1 + 117, i+«A
lY
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146
rummer's [50,
X.
g = Ae = 0; giving «' = y3,/3' = l+;3x 7=1 « + /3;
and the integral is
z^F^^, 1+^7, la+/3, J);
XI.
B = ^, e = ja0; giving £('=7 a, ^' = 1  a,
7' = 1  a + /3 ; on interchanging a' and ff, the
integral is
•(r"('
XII, Sa, e = 7a^; giving a'^^^, /3'=l/3,
7' = 1 + a — ^ ; on interchanging a' and ^', the
integral is
s'(l^J~^~^ f(i^, 7/3, l+«A J).
The remaining four sets, each containing four integrals, and
belonging to the substitutions
respectively, can be obtained in a similar manner*. Tliey are ; —
V. f(o, ft a + ,37+1, f);
VI. (lf)>^F<«7+l,/37 + l,<. + /37 + l, 0;
VII. fv«Ji'(7ii, 7ft 7a3 + l, 0;
VIII. (1  f)» f— « ^(la, lft 7«^ + l, f);
in which set £; denotes I —n:
XIII. fj?(«, 7A «;3+l, 0;
XIV. CF{/i, 7«, /3« + l. 0;
XV. (li;)"+>?*'(a7+l, 113, 113 + 1, ();
XVI. (lt)"»f'.f'(37+l. 1ci, /31I + 1, a;
in which set if denotes .j :
SVII. (lf)i?(», 7ft 7, f);
XVIII. (lflBfCft 7a, 7, 0;
* The complete set ot expreasions, differently obtained and originally due to
Knmmer, ate given in my Treatise on Differential EquaUom, (2nd ei.), pp. 192—
194; the Eoman numbers, used above to specify the cases, are in accord with the
numbere there used.
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50.] INTEGRALS 147
XIX. f^(if)'J?(a7 + i, :A 2,, O;
XX. i"'{li;}'F{/3i + l. l^c. 2y. 0;
in which set f denotes ; and
s — 1
XXI. (lf)"*'(«. <"7 + I. « + ^7 + l. E)i
XXn. (lf)»F(A^7 + l, a + /37 + l, t);
xxm. cy''0tyF(ia, ,,a, ia^ + i. Oi
XXIV. tr'(iO'r(ii3.j0. j~„i3 + i, 0;
in which set Jf denotes .
The preceding investigations have been based upon the assumption,
among others, that no one of the quantities
is an integer or aero : the determination ot the integrals of the differential
equation
when the assumption is not justified, can be effected by the methods of
§§ 36—38.
Consider, in particular, the int^rals in the vicinity of s = 0, when l~y is
an integer ; there are three cases, according as the int^er is zero, positive, or
negative. We substitute
».cy + «, .'+' + . ..+c..'+ + ...
in the equation ; and we find
zBto^e{e+yl)c„/,
provided
(^l +„ + fl)...(n + g)(wl + g + d)...(g + fl)
''" (>i + 6) {l + e){n. 1+7+5) (7 + ^) "■
(i) Let 1—7 = 0, so that the indiciil equation ls S^ = : then the two
integrals belong to the mdcs 0, and one of them certainly involves a
logarithm; and they aie t,nen by
«• m..
The former, when we take Cq—I, is
F(a, ft 1, z),
with the usual notation for the hypoi^eometric function ; as the coefficients
ai'e required for the other integral, we write
F{a,0, 1, s) = l+Ki2+«52^+... + =,.j"+....
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148 THE HYPEIIGEOMETRIC [50.
The second integral, when to it we add
C(, again being made ec[ual to unity, becomes
where ij/ (m) denotea ^ {log n (m)).
(ii) Let I — y be a positive integer, say p, where p>0. The indicial
equation, being S(S—p)=0, has its roots equal top, 0. We have
'^^ (n+e)...ii+6)(np+s)...(ip+e) '■
Of the two int^r .Is, that, which belongs to the greater of the two exponents,
is equal to
z''F{aip,^+p, i+p,z},
when we take <;,=0. The other integral may or may not involve logarithms.
If it is not to involve logarithms, then, as in § 41, the numerator of o^ must
vanish when 8=0, so that
(pl+a)...aipl + 0)...0
must vanish ; in other words, either a or 3 must be zero or a negative integer
not less than y. When this condition is satisfied, the integral belonging to
the index zero is e&ectively a polynomial in z of degree — a or  /3 aa the case
may be, and it contains a term independent of i.
When the preceding condition is not satisfled, the integral certainly
involves logarithms. In accordance with § 36, we take
Ga=ce,
so that
w = (7 S *■■
, (^l + a + g)...(a + fl) (»l +fl+g)...(g + g )
{n + 0)...{l + e) (np + e)...{lp + 0)'''
There are two integrals givOTi by
M [Sh
The first is easily seen to be a constant multiple of
z''F{a+p, 0+p, l+p, s),
thus in effect providing no new integral. The second, after redm
mating C^l, is
J> (»l+.)....(«H3)...3
+ „.»!(~y)(«lrt...(lj.)'
+(_i,.i; (— n.)..(»nw.g ,.
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50.]
=^(^_H.„)+^(«_i+3)>;,(«)f(»^^).
(iii) Let 1— 7 be a negative integer, say q, where 5>0. The indicial
equation, being ${S[q) =0, has its roots equal to 0,  q. We have
(«l+„+g)...(„+g)fal+g+g)...(3+g)
(™+6I),..(l+d){«+j+(9)... (1+3+19) ^'
The greater of the two exponents is 0; the integral which belongs to it, on
making Cf, = \, becomes
F{a,»,l¥q,z).
Tbfs integral which belongs to the exponent — g may, or may not, involve
logarithms. If it is not to involve logarithms, then, as before, the numerator
of tfj must vanish when ^ =  5, so that
(.l)...(a,;)((il)...(e,)
must vanish : hence either n or j9 must be a jxisitive integer greater than
and leas than y( = l + q). When the condition is satisfied, the jnt^ral is
a polynomial in s~\ beginning with £~', and ending with a~" or 3~P, as the
case may ba.
When the preceding condition is not satisfied, the integral certainly
involves logarithms. As before, in accordance with § 36, we take
e,~(»+})Jr,
... „. («~i+.H»)...(»+») ( «i+e+<)0K ),„i ,,,«H.
Two integrals are given by
The first is easily seen to be a constant multiple of
/■(n, 3,1+ J, 4
so that no new integral is thus provided. The second, after reduction, and
making K= 1, is
+(1,.. J (»lt g ).(°g)('l+gg)(8?) ,,— ,
*„=K«l+«?)+^(7iI+e?)v('(»i)i'(ns).
The integrals are thus obtained in all the cases, when y is an integer.
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150 EQUATIONS OF THE SECOND ORDER [50.
Siroilai' treatment can be applied to the integrala of the equ^ition, when
■ya— 3 i^ *" integer, positive, zero, or negative, contrary tg the original
hypothesis as to the exponents for 3=1 ; likewise, when a,8 is an integer,
positive, zero, or negative, contrary to the original hypothesis as to the
exponents for 3=a>. These instances are left as exercises.
Note. There is a great amount of literature dealing with the
hypergeometric series, with the linear equation which it satisfies,
and with the integrals of that equation. The detailed properties
of the series and all the associated series are of great importance :
bub as they are developed, they soon pass beyond the range of
illustrating the general theory of linear differential equations, and
become the special properties of the particular function. Accord
ingly, such properties will not here be discussed: they will be
found in Klein's lectures JJeher die hypergeometrische Function
(Gottingen, 1894), where many references to original authorities
will be found.
Equations of the Second Order and Fuchsian Type.
51. No equations of the Fuchsian type, other than those
already discussed, are made completely determinate merely by
the assignment of the singularities and their exponents. It is
expedient to consider one or two instances of equations, which
shall indicate how far they contain arbitrary elements after singu
larities and exponents are assigned.
Suppose that an equation of the second order has p singulari
ties in the finite part of the plane and has co for a singularity ;
the sum of the exponents which belong to these p + 1 singularities
is (by Ex. 2, § i6) equal to p — 1. Now let a homogiaphic substi
tution be applied to the independent variable and let it be chosen
so that all the points, congruent to the p+1 singularities, lie in
the finite part of the plane. Thus « is not a smgulaiity of the
transformed equation: there are p+1, say re, singulaiities in the
finite part of the plane: and the adopted transfoimation has not
affected the exponents, which accordingly aie transferied to the
respective congruent points. Hence, when an equation of the
second order and Fuchsian type has n singularities m the finite
part of the plane and when infinity is not a singularity, the sum
of the exponents belonging to the n points is equal to n — 2. For
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51.] AND FUCHSIAN TYPE 151
such an equation of the second order, let the singularities and
their exponents be
then
Let
^.VW = (»<..)(2a.).. .<««.);
then, as the equation is of the second order and as all its integrals
are regular, it is of the form
where F^ and F^ are polynomials in z of orders not higher than
n — \ and 2n — 2 respectively. Also, let
F _ A, A^ A„ .
■^ s—a, s — a^ '" s — a„ '
and let
F^ = F^(s)  A"e'^^ + B"2f^' + CV' + ....
The indicial equation for the point e = a,, is
'(o'^ + ^' + WmT"''
and therefore
a, + S, = l~A„
X A,. = n %(«, + 0,)
— '2,
and therefore the polynomial Fi is of the form
F, = 2s""'' + lower powers of z.
Again, X is to be an ordinary point of an integral ; hence, talcing
the most general case, we must have an integral
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152 EQUATIONS OF [51.
where K,, is not zero ; for otherwise we should have a special
limitation that every integral is zero at infinity. Substituting, so
as to have the equation identically satisfied, and writing
(so that Sa = 2), we find, as the necessary conditions,
= K,A",
= (2  s,) K, + K,A" + K^ \b" + 2^" 2 a\ ,
<i = {'l> 2s,\ A")K., + kJ s, + B" ^lA" i re,)
+ K„ \a"U I «,' + 2 2 ara)j + 2S" I a, + C'\ ,
and so on. The first gives
A" = 0;
then the second gives
both of these equations leaving Kg and Ki arbitrary. The third
equation then gives
and so on, in succession. The remaining coefficients K are
uniquely determinate; they are linear in Ki and K^, the various
coefficients involving the singularities and their exponents, as well
as the coefficients in F^. The equation therefore has z = <x,for
an ordinary point of its integrals, provided F^is of order not higher
than 2n — 4.
The equation can, in this case, be expressed in a different
form. Let
^= = 1(0"^* + ...)
= Pn4 + + ^ +,.. + — ^^ ,
sa, zas za^
where P,^^ is a polynomial of order n — i. (Of course, if 2n ~ 4
is less than n, which is the case when n = 3, there is no such
polynomial.) As the coefficients in F^ are not subject to any
further conditions in connection with the nature of 3= co for the
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51.] FUCHSIAN TYPE 153
integrals, any values or relations imposed upon \,, X^, .... >.„ and
the coefficients in Pnt must be associated with the singularities.
The equation now is
^„ / l»,ffA , 1/ _X, A _„^
The indicial equation for a = a, is
^ (fl  1) + (1  a,  ^.) ^ + ^,''^^— ^ = 0,
and its roots must be ctr, 0r thus
and therefore the equation is
It follows that the only coefficients which remain arbitrary are
the ji — 3 coefRcienta in the polynomial P,_ (where n ^ 4). When
the polynomial P„_i is arbitrarily taken, the foregoing is the
most general form of equation of the second order and of Fuchsian
type, which has n assigned singularities in the finite part of the
plane with assigned exponents, and has oo for an ordinary point of
its integrals. This is the form adopted by Klein*.
If a new dependent variable y be introduced, defined by the
relation
w = y {z — a^Y' {z  a^'"' ... (za^)'^,
then the exponents to which y belongs in the vicinity of a^ are
the difference of which is the same as for w ; but s = co will
have become a singularity, unless
Pi + Pi + ...+p^>0.
Now
^1 {(<.. ~ p.) + (0.  p,)) = n ~ 2  2 1^ />, ;
and therefore
i_ !1  (»,  p,)  (A  p,)i = 2 + 2 l^p,.
* VorlemngenUberlineareDifferentialgleichungendeiziueilenOrdnanglGottmgeu,
1894), p, 7.
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154 EQUATIONS OF [51.
Hence, if 3= ic is not to be a siDgularity, the quantities p,, ..., p^
cannot all be chosen so that each of the magnitudes
vanishes. Conversely, if the quantities p^ be chosen so that each
of these magnitudes vanishes, then z = oo has become a singularity
of the equation ; having regard to the form of w for large values
of 2, we see that and 1 are the exponents to which y belongs for
large values of z ; and the differential equation for y is easily seen
to be
where P„_3 is a polynomial of order n~S.
This equation, however, has n singularities in the finite part
of the plane, and a specially limited singularity at s = co : we
proceed, in the next paragraph, to the more genera! case.
Note. The indicia! equation for a = oo in the case of the
equation for w is
0(0 + l)0j^(la.^.) = O,
that is.
The root = gives an integral of the form
and the root = 1 gives an integral of the form
^(;*:
both of which are holomorphic for large values of s[, so that all
integrals are holomorphic functions of  for large values of \z\.
In this case, oo is not a singularity of the integrals : it can be
regarded as an apparent singularity of the differential equation,
and (if we please) we may consider and — 1 as its exponents.
£x. Shew that the preceding equation can be eshibited in the form
<!.^^^)t:.<
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51.] FUCHSIAN TYPE 155
where the n constanta c,, ..., o^ satisfy the three relations
2 c, = 0, S C^r+ 2 c^0r = O, 2 c^/ + 2 2 Orl3^ay = 0,
and otherwise are arbitrary in the most general case. (Klein.)
52. Now consider the equation of the second order and of
Fuchsian type, which has m singularities in the finite part of the
plane, say Oi, a^, ..., (X«, with exponents a, and /9i, ..., a„ and ^n,
respectively, and for which m also is a singularity with exponents
a and /3 : the exponents being subject to the relation
Let il' denote (z — a^)(sa^)...(s — an): then the equation is of
the form
w"+( I ^"l w' + ^^ w = 0,
where (? is a polynomial of order not higher than 2n  2. When
G is divided by yjr, we have a polynomial of order n2 and a
fractional part : and so we may write
The indicial equatioH for 3 = a^ now is
4,= la, ft,
holding for r = 1, 2, . . . , jj. The indicial equation for i
l)0 2 ^, + A„_, = 0,
X Ar1, /(„_ = a^;
the former being satisfied on account of the relation between the
exponents. The equation thus is
w" + 2 
.+;..+ s?^^!±>^>lo,
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156 A NORMAL FORM OF EQUATIONS [52.
the coefficients ht,, h^, ..., ^^,3 being independent of the singu
larities and their exponents.
When a new dependent variable y is defined by the trans
formation
w = (2 aO"" (^ «.)"»... (^  «„)"y,
then the exponents ofy for a, are and ^, — a,, say and X^, this
holding for r=l, 2, ,,., n: and its exponents for x are
BL+% a,, 0+i a,,,
= <r, T say : where
a + T+ X Xr = ii 1.
The function y is, in general character, similar to w: it has the
same singularities as w, and it is regular in the vicinity of each of
them but with altered exponents : and it thus satisfies an equa
tion of the second order and Fuchsian type, which (after the earlier
investigation) is
where i„_3, ..., k„ are independent of the singularities and their
exponents *.
This transformation of an equation
to an equation
„ , G., , ^ 0.. „
where Fai, F^_^, G^i, G„^2 are polynomials of order indicated
by their subscript index, appears to have been given first by
Fuchsf, The simplest example of importance occurs for n = 2,
when the hypergeometric equation is once more obtained.
53. It is well known that, when y is determined by the
equation
y" + Py'+Q = 0,
* The equation for y can be obtained by the direct substitution of the expreafiion
for w in tlie eailier differential equation for lu. When reduction takes plaee, there
t Heffter, Einleitung in die Tkeorie der Unearen DijferentialgUichungen, (1894),
p. 23i.
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63.] OF FUCHSIAN TYPE 157
and a new variable F is introdueed by tbo relation
the differential equation for ¥ is
where
i.eifjp..
In the case of the preceding equation, the relation between
y and Y is
SO that F is a regular integral in the vicinity of all the singular
ities and of t» , the exponents being
i(lXr), ^(1+Xr). fo5 s = .a„ ('■ = 1. ■.™).
and
■^ ( 1 + o  t), ^ ( 1  o + t), for s = iO .
From the form of P and Q, it is easy to see that
Ii^^ = polynomial of order 2w — 2
^ L ,.i^»J
where P„_2 is a polynomial of order )i — 2, say
P„  Bz— + (,_, 2" + . . . + i,.
In order that ^ (1 — V), ^ (1 + V) niay be the exponents of a^ for
the equation
Y" + IT = (1,
they must be the roots of
e(eV) + ,?'0:
hence
JJ,,.1(1V)^''(«,).
In order that ^ (— 1 + a — t), ^ (— 1 — ct + t) may be the exponents
of <Xi for the same differential equation, they must be the roots of
0(<f. + l) + C = O:
hence
Cill(.7^T)l.
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158 KLEIN'S NORMAL FORM [53.
The remaining' constants l^, l^, , Ins ^^^ expressible as homo
geneous linear functions of &„, i,, .,., k^^i, so that they are inde
pendent of the singularities and the exponents : and thus the
equation is
fJ+[lll()1" + !..^"+...+i.
r)].o.
Corollary. For the original equation, x was a singularity
of the integrals with exponents a and t. If it were only an
apparent singularity of the original equation, so that the integrals
are regular for lajge values of \s\, then we have the ease indicated
in the Note, § 51, so that we can take
<T, T = 0, 1.
The modified equation now is
For this differential equation and its integrals, the exponents to
which the integrals belong in the vicinity of Or are ^(1 — X,),
J(l+Xr); but 30 is now a singularity of the integrals, and the
exponents for a = co are 0, — 1, so that s = i» is a simple zero of
one of the linearly independent integrals of the modified equation.
dY
These forms of the equation, from which tho term in , is
absent, are the normal forms used by Klein.
The simpleBt example of the class of equations, not made entirely determ
inate by the assignment otthe singularities and their exponents, occurs when
there are three singularities in the finite part of the plane and oo also is a
singularity. By a homc^raphio transformation of the variahle, two of the
singularities can he made to occur at and 1, and cc can be left unaltered ;
let a denote the remaining singularity. Let the exponents he
0, 1X(,forz = 0; 0, lAifori=l; 0, X for j = u ;
o,rfor2=tt>;
where
crTX„Xi + X = 0.
Then the differential equation is
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53.] EXAMPLES 159
where q is the (sole) arbitrary constant, left undetermined by the assigned
properties. The integral of this equation, which is regular in the vioinitj of
3=0 and belongs to the index 0, is denoted* by
F{a, q; cr, t, Xj,, X,; s).
If «•=!, 3=1, the equation degenerates into that of a Gauss's hypergeometrie
series ; likewise if a = 0, j=0.
Ex. 1. Verify that, when a = ^, the group of substitutions
, i_, t ?ri !zi ^ ifci) i!_
'■ •■ i , • .V .!• .1 ' .i'
interchanges among themselves the four points 0, ^, 1, co .
Prove that, when a=~\ and when a =2, there is in each case a corre
sponding group of eigbt substitutions interchanging the points 0, 1, a, cd
among themselves: and that, when a=^{\+i^2) and when a=^{\i^3),
there is in each case a corresponding group of twelve substitutions. Construct
these groups. (Heun.)
Ex. 2. Prove that there are eight integrals of Heun's equation of tte
i^{z\f{za)yF{a,q; ,/, r', V. V 5 4
which are regular in the vicinity of the origin and have the same exponents
as F{a, q; a; T, X„, X, ; ?). Hence construct a set of 64 integrals for the
equation when «=J, which correspond to Kummer's set of 24 integrals for
the hypei^ometric series.
Indicate the corresponding results when
a=\, 3, ^(1 + W3), jai^a)
Ex. 3. A homogeneouiS linear differential equation of order n is to have n
singularities ts,, Qj, ,,., a^ in the finite part of the plane and also to have
00 for a singularity : the integrals are to be regular in the vicinity of each of
the singularities, and the exponents of 01^ are to be 0, 1, ,.., n — 2, a,, (for
c=l, ..., «), while the exponents of ro are to be 0, 1, ..., ra— 2, a, so that
a + J^«, = («l)^
Shew that the differential equation is
where ij» (s) = (s  Oj) (s  %). , .(j  o,), the coefiicient Eg (b) is a polynomial in z
of order not greater than », (for 3=1, ..., n), and
.E,(2)=£(a.«+l)^^.
(Pochh amm er. )
' Heun, Math. Ann., t. xxxiii (18891, PP. 161—179, who has developed some of
the properties of these equations, and has applied them, in another memoir fl.r,.,
pp. 180—196), to Lajnf s functions.
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160 equations op fuchsian type [54.
Equations in Mathematical Physics and Equations of
FucHsiAN Type.
54. These equations of Fuchsian type include many of the
differential equations of the second order that occur in mathe
matical physics; somebimes such an equation is explicitly of
Fuchsian type, sometimes it is a limiting form of an equation of
Fuchsian type.
One such example has already been indicated, in Legendre's
differential equation (Ex. 1, § 46). Another rises from a transform
ation of Lame's differential equation which ( 148) is of the form
1 d^ . , , 71 «
5J,+^S'W + B = (l.
whore A and B ai'O constants*. Writing
(•(,). c,,
so that ic is a new independent variable, we liave
d% / i , J i \diD , , Ax + B „
ejdx ' {X
The singularities of this equation are Bi, e^, e^, oo; the exponents
to which the integrals belong in the vicinity of Si, e^, e^ are and J,
in each case ; the exponents, to which they belong for large values
of a;, are the roots of the equation
The new equation is of Fuchsian type : and, in this form, it is
frequently called Lamp's equation.
An equation, similar to Lamp's equation, but having n singu
larities in the finite part of the plane, each of them with and J
as their exponents, as well as ^ = oo with exponents a and 0, such
that (§ 52)
«^^:=i«~l,
is sometimes called Lame's generalised equation. By § 52, it is of
the form
w" + w' i JI „'^"^° w = 0,
,^^z a^ Ji(^a,)
" This 13 the general iorni; tho value ~fi(K + l) is assigned (i.e.) to A, in
order to have those eases of the general form which possesa a uniform integral.
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5*.] IN MATHEMATICAL PHYSICS 161
where Gni is a polynomial of order n~2, the highest term in
which is a^s"~^.
55. The equation of Fuchsian type which, nest after the
equation determined by Riemann's Pfunctioii, appears to be of
moat interest is that for which there are five singularities in the
finite part of the plane, while ^ = x is an ordinary point. The
interest is caused by a theorem*, due to EScher, to the effect that
when the five points are made to coalesce in all possible ways, each
limiting form of the equation contains, or is equivalent to, one of the
linear equations of mathematical physics.
Let the points be a^, a^, (h, Wj, <h, with indices a^ and ^r, for
r=l, 2, 3,4, 5; then
and the equation (p. 153) is
where ^ = 11 (z — Or), and Pi is a linear polynomial Ax + B. The
substantially distinct modes of coalescence are :—
(i), ttj and ttj into one point ;
(ii), Ma and a^ into one point, a, and a,^ into another ;
(iii), tta, 0.1, 1X5 into one point ;
(iv), Oj and a^ into one point, ctj, at, Kj into another;
(v), at, Ob, ai, a, into one point ;
(vi), all five into one point ;
and the various cases will be considered in turn.
Gase (i). Let the indices for a^, a^, a^ be made 0, ^ for each
point ; then, as i/^' (0.4) = 0, 1^' (dj) = in the present case, and
* Ueber die Eeihenentwiokeluagen i!er Potentialthoorie, GStt. geirSnte Preis
sehri/t, (1891), p. 44; and a sepaiate took undpr the same title, p. 193. Sbb also
Klein, Varlesungtii iiber Uiieare DiJ/ircnhiljl itliuniie?i dee xxaeiten Ordmiiig, (1894),
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162 bocher's theorem on [55.
the equation is
* i F
Write
2 — »! =  , (ttr — «4) ^r = !> foi' '■ = 1, 2,3;
the equation becomes
d?wdw( i , \ . \ \ , C^ + I> ,.. ^
in effect, the preceding ungeneralised Lamp's equation.
Case (ii). The equation becomes
^.,^ ^,l a.ft ^ l^K^8 ' ^ l.'ri
+( .».)(.';).(.,.). {^"^/''''''4°''
itfber coalescence of the points, where
1a' jS' =2«,/3,«,/3„
1  a"  y9" = 2  a,  ^,  a,  ,3, ,
and therefore
fli + /3i + a' + ^' + a" + /3" = 1.
Writing = (z — a^{z — a.^{z — a,^, we have the coefficient of
^ in the form
za, (z a^) (z  ft.) '
where Q,, like P„ is an arbitrary linear polynomial. Thus Q,
contains two arbitrary coefficients ; these can be determined so
that
{z — at) {z — at) z — o^ ^  «j '
and then the equation becomes
.,/■ ^ .„■  i''.ft J. i«'g ^ i»"ri
«1 so, Jo, ao, J
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55.] EQUATIONS OF FUCH8IAN TYPE 163
Owing to the form of $ and the relation 2 (a + (S) = 1, this is the
equation of Eiemann's P function (§ 49).
When we write ai = l; (1^ = — 1; ai = <xi;
a„^, = 0, 0; a', ^' = 0,0; a", ^" = ~n,n+l ;
the equation becomes
that is,
{1  ^)w"  2zw' + n(n + l)w = 0,
which is Legendre's equation.
Case (iii). Let a, , A = 0, i ; a, , A = 0, i ; so that
1«,~A + 1 04^ + 1 «>ft = l.
After the coalescence of the points, the equation is
.„"..„.r_i_^.J
a,l (^o,)(F »,)(« ,..)■•
where P) is a linear polynomial, say {^ {z — as) + B}{a, — a^)(as — Oi).
Now let
after some easy reduction, the equation becomes
— , dta f 1 1
Us — Oj flj — Hj]
1
Let 01 = 00, (Xj — tt2 = — 1: the equation is
d'w , 2ai — 1 (^ A +
Writing a) = sin'' t, we have
^^w{A + B sin= = 0,
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164 LIMITING FORMS OF AN [55.
which is known* as the equation of tho elliptic cylinder. This
equation will be discussed hereafter {^ 138 — 140).
Case (iv). Let a„ A= 0, i ; '^.03=0, ^; so that, as in the
10,/3, + ! 0,^4+1 a,^, = l.
After coalescence of the points, the equation is
P,
 (hf'
Let
s«, = i P, = «(^«,) + /3}(a,«,)^ c{a,a,) = l;
then the equation becomes
dHv 1 f^w a + 00)
or, taking
we have
ci% 1 dw
dhu 1 dw /4a , ^\
ay" y dy \y^ J
which includes Bessel's equation, sometimes called the equation of
the circular cjKnder.
Case (v). Let «i, A = ", ^ ; then
2^(l«,^,) = .
and the equation, after coalescence of the points, becomes
«." + a.'(^ + L~) + ^ ?1 ^^„ = 0.
Let
2„, = i, P, = »(2o,) + ,31(a,o,). 6<»,o=)l;
then the equation is
dhu dw ^ a + ^x ^
da^ aw x — b x b
• Heine, Kutjelfunationen, t. i, p. 404.
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55.] EQUATION OF FUOHSIAN TYPE
Writing
the equation becomes
which is the equation* of the parabolic cylinder.
Case (vi). The equation is
w"+ w'+7 ^7W = 0:
when we take
the equation becomes
da?
This corresponds to no particular equation in mathematical
physics : it will be recognised as a very special instance of equa
tions most simply integrated by definite integralsf.
Ex. Discuss, in a aimilar manner, the limiting forms which are obtained
when the singularities of
(i) the equation determined by Riemann'a Pfunction,
(ii) Lamp's equation, expressed as an equation of Fuchsian type,
are made to coalesce in the various ways that are possible.
Equations with Integrals that are Polynomials,
56. There is one simple class of integrals which obey the
condition of being everywhere regular, so that their differential
equations are of the Fuchsian type ; it is the class constituted by
functions which are algebraic. We shall, however, reserve the
discussion of linear differential equations having algebraic inte
grals until the next chapter ; and we proceed to a brief dis
of a more limited question.
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166 EQUATIONS WITH [56.
We have seen that an equation of the second order and of
Fuchsian type can be transformed to
By = 1^" + (?„i/ + G„^y = 0.
Its integrals are regular in the vicinity of each of n singularities
and of infinity ; the question arises whether the coefficients in the
polynomials ©„_, and (?„_3 can be chosen so that one integral of
the equation at least shall be, not merely fi"ee from logarithms
or even algebraic, but actually a polynomial in s. This question
has been answered by Heine* ; the result is that &,^i can be
taken arbitrarily, and Q.^^ has then a limited number of determ
inations.
If the above equation, in which
G„_i = CoS"' + CiS"^ + . . . + C„_5Z + C^i ,
(?„_3 = k^^ + h^"^ + . . . + k^^ + ^„_2,
is satisfied by a polynomial of order ni, say by
y ^ g^"^ + gi^^'" + ■■• +9^1^ + 9,^,
then
80 that there are m + n — l relations among constants. The form
of these relations shews that gi, g^, ..., gm ^re multiples of ff,,' to
express these multiples, m of the relations are required, and when
the values obtained are substituted in the remainder, we have
n — 1 relations left, involving the constants c and k. Assuming
the points ai, a^, ..., a„ arbitrarily taken, and the coefficients
Co, Ci, .,., c„_] arbitrarily assigned, we shall have these n—1 rela
tions independent of one another, and therefore sufficient for the
determination of the K — 1 constants /c^, ^i, ...,^»_a.
The first of these relations is
m(ml) + o,m + k, = 0,
so that ks is uniquely determinate. Denoting by
[kuh. ...,K]r
the generic expression of a function of ki, k^, ..., k^, which is
polynomial in those quantities, and the terms of highest weight in
* Heine, Kugel/unctioneii, t. i, p. 473.
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56.] POLYNOMIAL I^fTEGRALS 167
which are of weight r, when weights 1, 2, . . . , u — 2 are assigned
to ^1,^2, ..., fcas, we have, from the m relations next after the
first,
for )■= 1, 2, .... m. When these are substituted in the remaining
w — 2 relations, we have
for s=l, 2, ..., « — 2. These determine the m — 2 constants
fci, ^2, ...,kn2', the number of determinations may be obtained
as follows. Writing
the equations become n  2 equations to determine w — 2 quantities
Xi, (c^, ,.., «Ba. In these quantities, the equations are of degrees
m + l,m + 2, ...,m+m2,
respectively; and therefore the number of sets of values for
iCi, x^, ..., Xn2 is
(m+l)(m + 2)...(m + «2).
But the same value of k^ is given by two values of x^, inde
pendently of the other constants k; so that the sets of values
of «i. a's, ■•, iTnamust range themselves in twos on this account.
Similarly, the same value of k^ is given by three values of iCj,
independently of the other constants k ; hence the arranged sets
of values must further range themselves in threes, on account of
kj. And so on, up to k^^. Hence, finally, the number of sets of
values of ^1, ..., kn^ is
(m + l)(m + 2)...(m + «2)
2.3...JI2
^ (ffi + H2)l
~ m 1 (n  2) I '
which therefore is the number of different quantities Gn~i per
mitting the equation
to possess* a polynomial integral of degree in.
* In Gonnection with these equationa, a memoir by Humbert, Jourw. de I'Ecole
Polytechaique, t. jliix (1880), pp. 207—220, may be consulted.
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168 POLYNOMIAL INTEGRALS [56.
This result is of importance, as being related to those special
forms of Lame's differential equation which possess an integral
expressible as a polynomial in an appropriate variable. This
polynomial can be taken as one of the regular integrals belonging
to each of the singularities ; the other regular integral belonging
to any singularity is, in general, a transcendental function and, in
general, it involves a logarithm in its expression.
Es!. I. Shew that a, linear equation of the third order, having all ita
integrals regular, caii, by appropriate transformation of its dependent variable,
be changed to the form
where
>/.=(^a,)(e<i,)...(E<i.),
Ill, itj, ..., On being all the singularities in the finite part of the iplane, and
■where P, Q, B are polynomial functions in z of degrees iil, 2ii2, 2ji— 3
leepectively.
Shew that, if P and § be arbitrarily chosen, R can be determined so that
one integral of the equation is a polynomial in i ; and prove that the number
of distinct values of B is
(m + 2»3) !
m\(,2nsy.'
where m is the degree of the polynomial integral.
Es. 2. Determine the conditions to be satisfied if
has two distinct polynomials as integrals, so that every integral is a poly
nomial.
Ex. 3. Determine how far the constants in the equation
may be assumed arbitrarily if the equation is to possess two polynomial
integrals.
Sc. i. Prove that the equation
/wg+l/'wt(.(.+ i)«*l»o
■where »i is an integer, f(x)~x^+a3fi+ba:+c, and a, b, c are constants, admits
of two integrals whose product is a polynomial in x.
Ex. 5, Shew that the only cases, in which the differential equation of the
.(i)2+(v(.+f.+i)«)*.»=o
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EXAMPLES 169
s whose product is a polynomial in ie of degree n, are as
follows. If n is an evea integer, then either a=^n; or j3=~Jn; or
a+/3=m, and y = ^, or ^, or , ..., or re+i. If ra is aJi odd integer,
then either a=\n and y=i, or , or f, ..,, or w+I, or ft or 3~1,
..., or »^\{n~\); or fi=\r<, and 7=^, or i, or f, ..., or i»i+l,
ora,oral, ...,or<.i(™l); or a4j3= «, and y=i, or i,or §,...,
or )i+^: (Markoff.)
Sr. 6. Sliew tliat, if the square root of a polynomial of degree m can be
an integral of tte equation
(V 2 Xrf^,)^'""^+''i^"
^+.
.+««2
n^(;^0
whore the exponents X and p. are subject to the usual relation, one of the
exponents \„ ^„ say X„ must be half of a noniiegative integer, this holding
for each value of s ; also ^ra — 2X, must be a nonnegative integer; and one
exponent of the singularity at iofinity must be equal to  ^»!.
If these conditions are satisfied, how many such equations exist?
n Vietk.)
Ex, 7. If the differential equation
"■" "•''"• 11 (».,)
where ^(^) is a polynomial, the constants a are real and positive, and tlie
Constanta e are real and distinct from one another, be satisfied by a poly
nomial ^ {x), then all the roots of (.c) are real, and no toot is leas than the
least or greater than the greatest of th.e quantities e. (Stieltjes ; BSoher.)
Equations with Rational Integrals,
■37. The investigation in § 56 suggests another question:
what are those linear equations, all the integrals of which are
rational naeromorphic functions of 3?
Let a,, ..., Oto he the singularities in the finite part of the
plane; let a^t, «^, ■•, a«r be the roots of the indicial equation for
Of, and let /3i, ..., 8n he the roots of the indicial equation for
z — x>. If every integral is to be a rational function of z, all the
roots a,r, Owi ■■, a„r must he integers; as no integral is to involve
a logarithm, no two of them may be equal. Let the arrangement
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170 EQUATIONS WITH [57.
of these roots be in decreasing order of the integers. The integral
belonging to the index a,r involves no logarithms ; in order that
the integrals belonging to the indices a„, a^r, ■■■, a»r respectively
may involve no logarithms,
H2+...+(nl),
that is, \n{'ii—l), conditions in all must be satisfied, these
conditions being as set out in § 41. Corresponding conditions
hold for each of the singularities, and also for «= « ; so that there
i»(«l)(m + l)
conditions of relation among the constants of the equation, in
addition to the necessity that the indicial equation of each singu
larity shall have unequal integers for its roots.
These conditions are certainly necessary ; they are also suffi
cient to secure that any integral of the equation is a rational
function of z. For considering the vicinity of a,, each integral in
that vicinity is of the form
where «„,. is the least of the roots of the indicial equation, and
Pm(3 — t,) is holomorphic in the vicinity of a,., for m= 1, ..., «;
when m = «, P{z —Of) does not vanish, and for all other values
of m it does vanish. If then o„r be zero or positive, the point
s = », is an ordinary point for every integral in the vicinity of «, ;
if Ojir be negative, then a,, is a pole of some integral, and it may be
a pole of several or of all.
As this holds in the vicinity of each of the singulaiities and of
s = 00 , it follows that, in the vicinity of every singularity of the
equation, including z — ca, every integral is uniform and has that
singularity either for an ordinary point or a pole ; moreover, every
integral is synectic in the vicinity of every other point : hence*
the integral is a rational function, which is a polynomial if oo be
the only pole. Thus the conditions are necessary and sufficient.
It has been seen that the indicial equation for each singularity
of the differential equation must have unequal integers for its
roots. When these are assigned arbitiarily, subject to the one
relation (Ex, 2, § 46) which they are bound to satisfy, they amount
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57.] RATIONAL INTEQEtAXS 171
to (m + 1) K  1 conditions ; so that the total number of necessary
conditions is
^n{nl){m + l) + (m + l)nl
= in(n + l)(m + l)l.
If such equations are being constructed, they are necessarily of
the form
where ^ = {z a^ ...{z~ a»), and 0^ is a polynomial of order not
greater than r{m— 1), for r= 1, ...,«. Henue the total number
of disposable constants is
m, from the positions of the singularities,
+ S \r{m\)+ 1}, from the constant coefficients in Gi, ..., G„,
that is,
i»(n + l)(ml) + » + ».
constants in all ; and therefore, in order that the equations may
exist, we must have
\n(n + \){m\) + n + m>\nin + X)(m + l)l,
so that
m^K^1.
In obtaining this result, an arbitrary assignment of unequal
integers as roots of the indicial equations has been made : and
it has been assumed that these conditions are independent of the
necessary conditions attaching to the coefficients, in order that
the integrals of the equation may be free from iogarithms. It
may, however, happen that a particular assignment does not leave
all these conditions independent of one another, so that we might
have
i»(» + l)(».l)+« + m.in(»+l)(m+l)l\,
and therefore
m = 7i'l\,
and still have the equation determinate. An instance is furnished
by the equation
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172 EQUATIONS WITH [57.
which, although it has only one singularity in the finite part of
the plane, so that m = 1, i! = 2, has an integral Ax"^ + Bx. For the
most general case, however, we have
JSr. 1. Investigate all the oasea in which the differential equation of the
hypergQometric series haa every integral a I'ational function of the independ
ent variable.
fiiir. 2. When the equation is of the second order, and all the assignments
of integer roots are quite general, the smallest value of m is 3. Let the
singularities be «[, ..., %,, with exponents aj, S,; a^, 9ji ...; a^, ft„; and
let the exponent* for ^= a> be a, S Choosing in each case the smaller of the
two indices Or and (9r, let it he o,., for r = l, ..,,nt; then writing
\r = »,ar, a+ S n^ = <r, 3+ S o^=r,
we have (§ 52)
cr + r+ S \^m\,
which is the necessary relation among the exponents. Writing
so that y also is a rational ftioction of z, our equation m y becomes
say
and here the integers Xj, X^, ,,., X,„ are, each of tliem, equal to or greater than
Substituting, in the vicinity of a^, the expression
(ia.)^Dy=c,0{e^K)A
provided
c„(C+«)(d + mX,.)Hc„
g(«,)
and the summation for a is for s=l, ..., m except s—r. As X^ is a positive
integer, and thus is the greater root of tlie modified indicial equation, there is
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57.] RATIONAL INTEGRALS 173
one regular integral belonging to the exponent X^, whicii is a constant multiple
of
= y, say, where y^ = c^^Cff, when 5 = A,,
When we write
and solve the equations for v^, o.^, ..., we find
K{&)
"' /(e+i).../(d+ur"
We know (§ 41) that there is a single condition to be satisfied in order that
the integral belonging to the exponent may be free from logarithms ; as
f(,6+n) vanishes to the first order for 6=0 when w=Xr, the condition is
There i^ i jrrespundmg cond tion fjr eaih of the sm^ulirities ajii foi 4, = =o ,
s) thit we haie in + 1 condit una, wtiih miohe the aibitrary constants
;! ^n 3 ind t.he positions of the singularities ds well as the assigned
integers \ ^^ a t Keeping the latter arlitrary, we aee that there
m Lst 1 e at least three singularities in the finite part ot the plane when
there are onlj three we oltiin a limited number of determinations of the
equation it there aie ''+jo then p elements are left aibitrary among an
otherwise limited number of determinatiins Df the equation*
As the oquition is of the setond eider it is possible to plotted otherwise
Assuming that the integial J" which bekngs to the exponent A^ f the
singularity a^ la known, and denoting by ^the mtegiil whuh 1 elongs tc the
exponent cf the same singulaiity welia\e
rz" Y"z+{yz'rz) 2 i^=o,
so that
and therefore
d (Z\ , 1 ™ , ,i 1
When the righthand side is expanded in powers of s—a^, the first term
involves {e—ar)~^~^', that is, the indes is negative. If ^ is to be free from
Ic^arithms, the term in in this expansion must have its coefficient equal
to zero — a condition which must be the equivalent of
• The hypergeometric case indicated in the preceding example is given by
»,.x.....=x„i, a. ,,[,„,). ..if„j.
which will be found to satisfy the conditions for a,, ... , a„ given in the text.
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CHAPTER V.
Linear Equations of the Second and the Third Orders
POSSESSING Algebraic Integrals.
58. The general form of equation, having all its integrals
regular in the vicinity of each of the singularities (including oo ),
has been obtained ; in the vicinity of a singularity a, each such
integral is of the form
(,^arH>,+ 'hiog{.~a) + i,,\kgUa)]' + ...+4,.\hg{za:)]l
where each of the functions ^n, 0,, ..., 0, is holomorphic at and
near a. In general, each of the functions is a transcendental
function in the domain of a: they are polynomials only when
special relations among the coefficients are satisfied.
When attention is paid to the aggregate of the integrals so
obtained, it is to be noted that the branches of a function defined
by means of an algebraic equation belong to this class. If
algebraic functions are to be integrals of the differential equation,
they constitute a special class ; special relations among coefficients
of the differential equation must then be satisfied, and, it may be,
special restrictions must be imposed upon its form. Accordingly,
we proceed to consider those linear equations whose integrals are
algebraic functions, that is, functions of s defined by an algebraic
equation between w; and z. It has already been proved (§ 17)
that each root of such an algebraic equation of any degree in iv
satisfies a homogeneous linear differential equation, the coefficients
of which are rational functions of z. If the algebraic equation
were resoluble into a number of other algebraic equations, neces
sarily of lower degree, each such component equation would lead
to its own differential equation of correspondingly lower order;
accordingly, we shall assume that the algebraic equation is irre
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58.] LINEAR SUBSTITUTIONS 175
soluble and proceed to consider linear ditFerential equations whose
integrals are tiie roots of an algebraic equation, , In the most
genera! case, the degree of the algebraic equation is equal to the
order of the diffei'entiai equation ; in particular cases ( 17, Note 1)
it can be greater than the order : and as we seek algebraic inte
grals, it may be expected that these particular cases will occur.
The investigation can be connected with an equivalent problem
that arises in a different range of ideas. It has been proved that,
given a fundamental system Wi, w^, ..., w^ of integrals of a linear
equation of order m,, the effect upon the system, caused by the
desciiption of a closed path enclosing one or more of the singu
larities, is to replace the system by another of the form
Wm = OmlWi + Oj^W.^ + ... + 0,„mWm I
say
(iU]' Wm') — S{Wi, ...,w,„),
where S denotes a linear substitution. By making the inde
pendent variable describe an unlimited number of contours any
number of times, we may obtain an unlimited number of linear
substitutions ; and so each integral could, in that case, be
made to have an unlimited number of values. If, however, the
fundamental system is equivalent to the m roots of an algebraic
equation, then each of the integrals can acquire only a limited
number of values at a point which are distinct from one another:
that is, there can be only a limited number of substitutions in
the aggregate. When therefore we know all the groups of linear
substitutions in m variables which are of finite order, only those
linear differential equations which possess such groups need be
considered. Accordingly, if we proceed by this method, it is
necessary to construct the finite groups of linear substitutions.
Further, it is clear that the investigation can be associated
with the theory of invariantive forms ; for the relations between
w/, ..., Wju' and w,, ..., w^ constitute a linear transformation of
the type under which these invariantive forms persist. Indeed,
it was by this association with binary, ternary, and quaternary
forms that the earliest results, relating to linear equations of the
orders two, three, and four, were obtained. Some brief indications
of this method will be given later (^ 69 — 72).
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EQUATIONS OF THE SECOND ORDER [59,
Klein's Method tor Equations of the Second Order.
59. The determination of linear equations of the second order,
whose integrals are everywhere algebraic, is effected by Klein*, by
a special method that associates it with the finite groups of linear
substitutions of two homogeneous variables.
Let w, and w, denote a fundamental system of integrals for
the differential equation ; and let
Ifi = awi + jSws, JTs = 7W, + Swj ,
be any one of the linear substitutions, representing the change
made upon the fundamental system by the description of a closed
path. Then taking
Wj'
the quotient of two algebraic integrals, so that s itself is an
algebraic function, we have
W.^ys + B'
thus s is subject to a homographic substitution. Accordingly,
the determination of the finite groups of linear substitutions in
the present case is effectively the determination of the finite
groups of homographic substitutions.
Let any such group containing N substitutions be represented
by
t.W. t.(«) +«(«).
and let t^,, (s) = s, the identical substitution : every possible com
bination of these substitutions can be expressed as some one of
the members of the group. Take a couple of arbitrary constants
a and b, subject solely to the negative restrictions that a is not
equal to ■<frr(b) and b is not equal to ^((a), for any of the values
0, 1, .,., N ~1 of r and of s; and form the equation
^a(s) — a ifrj (s)  a t^tj^., (s )  a _ „
Ms)~b■^ir,{s)b ir^d'^)b
* Math. Ann., t. ii (1877), pp. 115—110, ih., t. xu (1877), pp. 107—179;
Varlemngen Hberdai Ikosaeder, {LeipKig, Teubaer, 1S84), pp. 113—123.
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59.] wrrn algebraic integrals 177
which is an algebraic equation of degree W^ in s. It is unaltered
when s is submitted to any of the substitutions of the group ; for
such a substitution only effects a permutation of the various N
fractions on the lefthand side among one another. Hence, if any
root s be known, all the N roots caa be derived from it by
submitting it to the JV substitutions of the group in turn.
For quite general values of X, the iV roots of the equation are
distinct; but it can happen that, for particular values of X, a
repeated root arises, of multiplicity v. From the nature of the
equation in relation to the group of substitutions, it follows that
each distinct root is of multiplicity c, so that there are N—v
distinct roots. To consider the effect of this property of the
equation, let the latter be changed so that the numerator and
denominator are mulfcipKed by the denominators of i'i(s), ...,
^fr^^l(s). It thus Can be expressed in the form
where G (s, a) is a polynomial in s of degree N", the coefficients
being functions of a, and G {s, J) is a similar polynomial, its
coefficients being the same functions of b. Let X, be a value
of X, such that 5 = <t, is a root of multiplicity v^ when X = X^ ;
then the equation
G(s,a) 0{a,,a) _
Q\s, h) G(cr„ b) '
N
has — roots each of multiplicity v^ when X =X,. But each such
root is a root of multiplicity iij ~ 1 of the equation
d [ gfoo) gfa, ■») )_■
'<u\e{s.bj G(r„6)( ■
that is, of the equation
AW = fl(»..)^«it^<J(..)«(''A) = 0;
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178 Klein's method for [59.
of fclie roots of this derived equation. Moreover, we then have
G(s,b}G{<y„b)
= XX,.
Let X^ be another vahie of X, suoh that s = cr^ is a root of the
equation of multiplicity jj when X = X^. A precisely similar
argument shews that each distinct root of the equation is of
multiplicity v^; that there are N^v^ distinct roots; that each
such root is of multiplicity v^—l for the equation A (s) = ; that
these roots account for
f(.i)
of the roots of the derived equation ; and that we have
O/' r x
N
where 'I's is a polynomial in s of deg;ree ■— .
Proceeding in this way with the various values of X that lead
to multiple roots of the initial equation, we shall exhaust all the
roots of the equation A (s) = 0. The degree of A (s) is 2jV — 2 ;
for if
G(s.a) = s^Ma) + s'''Ma)+ ...,
then
B{s.b)^/.(b) + s'''Mb)+...;
and therefore
4(.)»'"l/.(«)/>(6)/.(4)/.(«)) + .
But taking account of the roots of A (s) = 0, as associated with the
multiple roots of the original equation for the respective values of
X, we see that its d
and therefore
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59.] EQUATIONS OF THE SECOND OKDEIi 179
Each of the integers v is equal to or greater than 2, so that each
of the quantities 1 is equal to or greater than J. Hence the
smallest number of different integers i is two ; if there were only
one, the lefthand side would be < 1, while the righthand side is
> 1. The largest number of different integers p is three ; if there
were four or more, the lefthand side would be equal to or greater
than 2, while the righthand side is less than 2.
In the first place, let there be only two integers, y, and v^ ;
then
1 J. __2
From the nature of the case, v, < iV, v^ ^ JV, so that
hence the only possible solution is
:,. = JV, v, = N; (I),
and N is an undetermined integer.
In the next place, let there be three integers, v^, v^, v, : then
111,2
Vi Vi 13 JS
At least one of the integers v must be 2 : for if each of these
integers were ^ 3, the lefthand side would be < 1, while the
righthand side is >1, as iV is a finite integer.
Taking j',= 2, we have
11,2
Another of the integers v may be 2. Let it be v^\ then N = ^V3,
and we have the solution
v, = 2, v,^2, v, = n, JV=27i, (II),
where n is an undetermined integer.
If neither of the integers p^ and v^ be 2, one of them
must be 3 ; for if each of them were ^ 4, then  + —^^, and so
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180 ALGEBRAIC INTEGBALS AND [59.
could certainly not be equal to ^ + ^ . Taking v^ — 3, we have
jV
N'
so that Cj < 6 : thus possible values of v^ aie 3, 4, 5. The solu
tions are
j,, = 2, r, = 3, v^=% N=12, (Ill),
„j=2, ^3 = 3, i; = i, N=2i, (IV),
v, = 2, v^^S, v,= 5, iV=60 (V).
60. The finite groups are thus known ; the corresponding
equations in s are required. The solutions will be taken in
order.
I. Instead of X, we take a quantity Z, defined by the rela
tion
, xx,
so that Z=0 gives X — X^, that is, gives s = Si, a root repeated
N times, and Z—xi gives X = X^, that is, gives s = s^, a root
repeated N times. We have
J V (''■)••
' e(s, l>)G(j„i)'
XX,~^i^,:
and therefore
absorbing the constant (? (Sj, b)rG (s^, b) into the variable Z.
11, III, IV, V. These cases are of the same general form.
Instead of X, we take a quantity Z, defined by the relation
XX, X,X,'
then 7=0 gives j: = X„ 2.1 gives X = X,, 2=« givesXX,,
and thus
Z:Z1 : 1
= (X  X,) (X,  X.) : (X  X,) (X,  X.) : (X  X.) (X,  X,).
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60.] POLYHEDRAL FUNCTIONS 181
But
' G(s,b)G(aub)'
Y Y  '^°^'
^ G{s,b)G(^,,b)'
^~^'^G{s,b)G(^,,by
and therefore
^ : 21 : 1 = ^*/^(s) : B<t','(s) : ^."'(s).
where A and B are constants which, if we please, may be absorbed
into the functions 3>s and <I», respectively.
Now these groups are the groups that occur in connection
with the polyhedral functions* : and the polyhedral functions can
be associated with the conformal repreaentation, upon a halfplane,
of a triangle, bounded by three circular arcs and having angles
equal to  ,  ,  , The analytical results connected with these
investigations can be at once applied to the present problem.
Denoting derivatives of Z with regard to s by Z', Z" , Z"', ..., we
have (T. F., § 275)
Z'\Z' ^'\Z')\ ^
or, taking account of the properties^ of the Schwarzian derivative,
we have
' ■ ' Z '^ (21)" Z{Zl)
The forms of the functions for the various cases II, III, IV, V
for II,
* T. P„ g§ 276—279, 300—302.
t T. F., %% 27*. 275.
+ See Ex. 3, § 62, of my Treatise on Differential Equations.
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182 ALGEBRAIC INTEGRALS OF [60.
for III,
Z:Z\ :1
= (s* + 2sV3  1)= : lav's sHs* + 1)^ : (s^  2s^V3  1)= ;
for IV,
Z:Z~l :1
= (s^ + 14s* + l)»:(s^=33s'33s'+l)^: 1085^(5' 1)= ;
and for V,
Z.Z1 : 1 = (s^~~ 228s" + 49is"' + 228s' + If
: {s*=+ 1 + 522s''(s™  1)  10005s'Hs"' + l)p
;1728s«(s" + lls°l)=.
These results* can be obtained by purely algebraic processes,
from the properties of finite groups proved by Gordanf.
61. These results can be applied at once to the determination
of linear equations of the second order
<Pw dw
all the integrals of which are algebraic. Denoting the quotient
of two integrals Wi and w^ by s, we have§
w =5'"*se"*^'^ w =s''^e'^'^'^, w.,s = w
say. As all integrals are to be algebraic, it follows that s and
s'S are algebraic ; accordingly, fpdz must be the logarithm of an
algebraic fv/nction, which is a Jirst condition. Further, in the
equations under consideration, both p and q (and therefore also
2/) are rational functions of z ; and therefore
[s, z] = rational function of z,
* They are sliglitly changed front the forms in % 302, g 278 {I.e.) ; the ciiange is
made, eo aa to associate the indices v.^, t^, y.j with the values Z = 0, Z — 1. Z = rrj
respeo lively.
t Math. Ann., t. m (1877), pp. 23^6. See also Cajley's memoir, "On the
achwaraian derivative and the polyhedral functions," Coll. Math. Papers, t. Si,
pp. 148—216.
g See my Treatise on Differential Equaiions, §g SI, G2.
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61.] EQUATIONS OF THE SECOND ORDER 183
and the quantity s is subject to the transformation of the finite
group. Now we have seen that
\'.^ F~+ ikif + Wz^ '
in cases II, III, IV, V ; and for case I, it is easy to verify directly
that
From the properties of the Schwarzian derivative, we have
hence, taking account of the particular form of [s, Z] which is
actually known, and of the generic form of [s, z] which ia required,
we see that, in order to satisfy the conditions, we must have
Z=R{z),
where K is a rational function of z. Conversely, the conditions will
I if .Z' is any rational function of z. Accordingly, the
<,l equation oftJie second order viust have the coefficient of
It/ in the form
where u is om algebraic function of z ; and its invariant I{z),
which is q — \P' — h'j^' '"''"^* ^s of the form
1. — + ^*^
^{Z\Y^ Z{Z^i)
\\Z,z},
where Z is any rational function of z ; the integers v^, v^, va in the
first form are the integers of the finite groups in cases II, III, IV,
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184 MODE OF OBTAINING [61.
V ; and N in the second form is an integer. When these con
ditions are Batisfied, the integrals are given by
where, for the first form, s is determined in terms of Z, the
rational function of z, by the equations at the end of § 60 ; and for
the second form.
Construction of an Integral, when it is Algebeaic.
62. The preceding investigation is adequate for the general
construction of linear equations of the second order which are
integrable algebraically ; there still remains the question of
determining whether any particular given equation satisfies
the test.
When the equation is of the form
d'vj dw
inspection of the form of p at once determines whether it satisfies
the condition which governs it specially. Assuming this con
dition to be satisfied, we construct the invariant I{z) of the
equation, where
and then, if the original equation is algebraically integrable, we
must also have
(21)' Z{Z1)
/Wl^^TT, +11^.^
©■il^^!.
where ^ is a rational function of 2, and the integers c,, v^, vs
belong to one of four definite systems.
It may happen that the identification is easy, because Z has
some simple value; the simplest of all is, of course, given by
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62.] ALGEBRAIC INTEGEAL9 185
Z= z. When the identification is not thus obvious, it is desirable
to have a method of constnicting the rational function Z if it
exists ; when it has been constructed, the further id entiii cation is
only a matter of comparing coefficients. Should this identification
be completely effected, then the integration of the equation is
given by the results of § 60,
Such a method is given by Klein*, who uses for the purpose a
comparison of those terms on the two sides, which are connected
with the poles and have the highest negative index. A rational
function is detenninate save as to a constant factor, when its
zeros, its poles in the iinite part of the plane, and their respective
multiplicities, all are known ; and this constant factor is determ
inate, when the value of the rational function is known for any
other value of the variable. Accordingly, let a denote a zero of ^
of multiplicity a, and so for all the zeros ; let c denote a pole of Z
{and therefore also of Z— 1) of multiplicity y, and so for all the
poles ; and let h denote a zero ot Z — 1 of multiplicity ^, and so
for all its zeros : then
U{zay U{bc)y
n{b~ay ■ n(zc)y'
where the multiplicity ^ of 6 is not used directly in the ex
pression.
Consider now the righthand side of the expression for I (i).
In the vicinity of a, we have
where t7 is a regular function oi 2~a, not vanisiiing wlien s = a;
so tliat
IdZ 'J „, .
^JJ^3i + '*<'<■>.
and
the unexpressed terms in [Z, z] having exponents greater than — 2.
In the vicinity of c, we have
Z=(zcyiV, Zl=(zc)iV,.
" Math. Aim., t. sii (1877), pp. 173—176; the espoaitiou given in the test
does not follow his eiactlj, as he transforms the equation 90 aa to secure tliat
s = o: ia an ordinary point.
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186 CONSTRUCTION OF AK [62.
where V and V^ are regular functions of 2 — c, not vanishing when
z=c; thus
IdZ 7 „,
1_ dZ ,
Zldz ^ z
_fS,{^c),
1^. ^
e)" ■■■'
the unexpressed terms in \Z, s] having exponents greater than — 2.
In the vicinity of 6, we have
2l(»6)»If,
where TT is a regular function oi s — h, not vanishing when z=h\
Ho that
Z\d^ zb* ^ °''
the unexpressed terms in [Z, z\ having exponents greater than — 2.
We thus have taken account of all the highest terras with
negative indices which arise through zeros or poles of Z and Z—1.
On account of the form of [Z, z], which is
Z' ^\Z') '
it is necessary to take account of the poles and the zeros of Z'.
As Z is rational, all its poles are poles of Z' and the latter has no
others; so that, on this score, no new terms arise, A repeated
zero of ^ is a zero of Z', and all these have heen taken into
account; likewise for a repeated zero of Z—1. Hence we need
only consider those roots of Z', which are not repeated roots of Z
or of if — 1 ; let such an one be t, of multiplicity t, so that
z.(ztYque,.
where Q is a regular function of ^ — (, not vanishing when s = t;
then
the unexpressed terms io [Z, s] having exponents greater than — 2.
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62.] ALGEBRAIC INTEGRAL 187
Gathering together the terms with the largest negative index,
we have, for Cases II, III, IV, V,
(s ay (s  by {z  cf {z  ty
where the unexpressed terms have integer exponents greater
than — 2 ; and in this expression the significance of a, b, c, for the
construction of Z, must be borne in mind. Actual comparison
with the form of / {z) then gives indications as to which set of
values of II,, Vi, Vj must he chosen, and determines the values of
a, ^, 7 The construction of Z is then possible and, Z being known,
the complete identification of the righthand side with the known
value of /(if) is merely a matter of numerical calculation.
For Case I, we have
and the method of proceeding is the same as before.
In particular instances, it may happen that no terms of the
type
tr + JT
(stf
occur : Z' then contains no roots other than the repeated roots of
Z and Z — 1. An example is given by
^ 4^'
Further, it may happen that a= v^, or ^ = vi, or y^Vsi so
that the corresponding value of z, viz. a, b, or c, is then not a
singularity of the differential equation. And, in particular, if
if = CO is not a singularity of the differential equation and there
fore also not a singularity of the integral, then, if the equation be
integrable algebraically, the numerator of the rational function Z
is a polynomial in z of the same degree as the denominator*.
" Thin form of equation is discoased by Klein in the memoir already quoted
(note, p. 186) ; reference should he made to it tor further developments.
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188
M:^. 1. The equatio]
blo algebraically, For
l% = l(l^)> whence J.2 = 2;
We thus have an instance of case II, when a = 2. All the conditions a
satisfied : and thus (§ 60) the integrals of the equation are givon by
fie. 3. Construct a linear differential equation of the second order in its
normal form, such that the quotient s of two of its solutions is given by
108s»(sil)a 43 '
Sx. 3. Consider the equation
We have
. ^ 2s>8s=153a82 + 2 (al) '
the terms indicated constituting all the infinities of /(s) of the second order.
First, it is clear that there is only one root of Z' other than repeated roots
of Z and Z— 1 ; it is characterised by
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62.] Klein's method 189
If it wereposBibly an instance of case II with m = 3, then we must have
^^l_^^^=^,sothati., = 2, 3 = 1, h^i,
i{^^^^> >',3,y= I, <: = (>,
i(l5) = 3^. ., = 3,y = 2,«=l;
and therefore
with the condition that 2—1 when z = h = i, so that A=i. But then
shewing that Z' docs not possess a root z = l= 1 ; hence the example is not an
instance of case II.
If therefore the equation is algebraically integrahle, it must bo an instance
of case III. We must have therefore
i{ 1 — ^)=A^i whence ff=l, b=i,
and then, either
Q = l, a = Q, y = % c=l;
Taking the former, we have
from the poles and zeros ai Z\ s& Z= 1, when s = i, we have A = 2, bo that
SO that Z I has the roots 2 = 8, s=  i; hut
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shewing that Z' does not possess the f.
values is not possible.
Taking the latter, we have
; and thus the first assignment of
■sof^; asZ=l when si,
e have A =^, and then
so that Z—\ has 2=i,z— ifor roots, and Z" has 2=1 for a root.
The preliroinavy conditions are thus satisfied ; it is easy to verify that
this value of Z gives the coinpleto value of /(:). Hence, after the results of
§ 60, the intf^ral of the differential equation is given by the equations
^si + 2aV3l Y _ {aVVf
\} 2s '
algebraically integrable.
^_ +<Si*.)sAo,
and bt n th ntegrals
where f=(iai)(^a2)(^''s),^dX,=i{aa),X,=iO^),),3=My/}i
discuss the possibilities of algebraic integrability for the values
\^h ^2 = f. \ = \
In particular, shew that, if %=  1, «3=0, then
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EQUATIONS OF THE THiRD ORDER
Equations of the Thied Order with Algebraic Integrals.
63, When we pass to the considGration of hnear equations of
order higher than the second which are algebraically integrable,
the discussion can be initiated in the same way as for equations
of the second order ; but the detailed devdopment proves to be
exceedingly laborious, and it has not been fully completed for
each case. Only a sketch will here be given
Dealing in particular with the linear equation of the third
order, we take it in the form
where p, q, r are rational functions of s, subject to the limitations
imposed by the regularity of the integrals in the vicinity of eafih
singularity (x included). If w^, w^, w, denote three i
independent integrals, we have (§ 9)
BO that, as Wj, Wa, Ws are algebraic functions of s, it follows that p,
a rational function of z, must be of the form
where m is an algebraic function of s. This is a first condition : it
is the same as for the equation of the second order (§ 61): and it
is easily obtained as a universal condition attaching to any linear
equation which is algebraically integrable.
Now substitute for v) by the relation
and let y^, y^, y^ denote the three integrals corresponding to
M>i, Wa, Misi owing to the character of p and the functional
character of the integrals w, the integrals y are also algebraic
functions of z. Thus the equation in y, being
/" + 3Qy + ii0,
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192 EQUATIONS OF THE THIRD ORDER [63.
where
Q = qp'p' I
R ^ r  Spq i ^p'  p" ) '
is to be algebraically integrable. Denoting by s and t the
quotients of two integrals by a third, we have
The quantities s and t are algebraic functions of z for equations
of the class under consideration.
The effect upon a fundamental system, when the independent
variable describes a circuit enclosing one or more of the singulari
ties, is represented by relations of the form
F, = a y, + b j/a + c
Fj = «' ^1 + b' y^ + c'
7a = (t"l/i + &>, + c"(/3 J
If S and T denote the corresponding integralquotients, then
„ ^ a' + b's + c't „ ^ a" + b"s + o "t
a + bs + ci ' a + bs + ct
Now if the equation is integrable algebraically, there can exist
only a limited number of different sets of values of the integrals ;
so that the number of sets Y,, ¥^, Y^ is finite, and the number of
simultaneous values of S and T is finite. If then we know all the
homogeneous linear gioups m three variables, or (what is the
same thing) all the lineolineai groups in two variables, which are
finite, then each such finite group determines its set of values of
Yi, Y,, Fa and the set of values of S and T, and so it determines
a linear equation the integrals of which are algebraic: and con
versely, each such linear equation is characterised by a finite
group.
64. In order to utilise the method for the present purpose
on the lines adopted for the equation of the second order, it is
necessary to deduce from the differential equation certain differen
tial invariants involving s and *, these invariants being expressed
in terms of Q and E. This can be done in two ways. It is clear
that, as s implicitly contains five arbitrary constants, it satisfies a
differentia! equation of order five ; and that, as ( is of the same
functional form as s, it satisfies the same differential equation.
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64.] WITH ALGEBRAIC INTEGRALS 193
On the other hand, as s and ( combined contain eight arbitrary
constants implicitly, it may be expected that the two differential
equations, which they satisfy and which wili involve both of them,
will be each of the fourth order or will he equivalent to two of
the fourth order. The single equation is, for some purposes, the
more important in the formal theory of the hnear equation, which
will be left undiscussed ; for the present purpose, the two equa
tions prove to be the more important. Accordingly, we substitute
sy, for 1/2, and tz/, for y^,
1 the equation
I integral of this equation, >
whence, remembering that y^
have
3s'y," + 3fi'>/ + {SQs' + s'") y, = 0
3(>," + 2t'%' + (SQt' + t'") !/, = 0)
Differentiating each of these once, and substituting for y,'" from
the linear equation which it satisfies, we have
Qs'%" + (4s'" ~ 6Qs') y/ + [s"" + SQs" + 3 (Q'  R) s'] 2/1 = 0)
Qt"y," + (W"  GQt') y: + [*"" + 3Qt" + 3 (Q'  ii) t'} y, = 0\'
so that there are four equations, linear and homogeneous in the
quantities y", y[, y,. When the ratios of y" : yl : y^ are eliminated
from the first pair and the first of the second pair, we have
3(iiQ')
and when the same ratios a
pair and the second of the se
i likewise eliminated from the first
>nd pair, we have
("", «'"
6!"
3Q
»'", 3."
3«'
1'", 3i"
31'
3(EQ')
('", 3f", 3('
These, in fact, are the two equatio
satisfied by s and (.
F. IV.
0.
f , ,
s"\ 35", 3s'
*"', 3(", 3('
, each of the fourth order,
13
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194 INVARIANTS FOE AN EQUATION [64
Suppose now that two solutions (other than the trivial solu
tions, s = constant, ( = constant) are known, say
Solving the first pair of the foregoing equations for y^ : y,,
we have
3 (aW  oV") y,' + (^'"t  a'r'") y, = 0,
and therefore
neglecting an arhitrary constant arising as a factor on the right
hand side. Hence a fundamental system of integrals of the
original equation is
{</'t
n~
tWt'c
■i.
or the original equation can be integrated if two particular
solutions of the equations in s and t are known.
65, Moreover, from the source of the two equations which serve
to determine s and i, it is to be expected that, when the above
two (being any two) particular solutions s— a, t = T, are known,
the complete primitive of the two equations is
'/ + b'lT + c't
ll+b(T+ GT
t =
where the constants a, h, c, a', V, c', a", b", c" are arbitrary so fe,r
as those two equations are concerned. This result can be stated
in a different form. The two equations in question can be written
As"" + iBs"' + 6Gs"  SQ (As" + 2Bs')  3 (K  Q') As' = 0,
Alf'" + 4Bt"' + 6Gt"  SQ (At" + 2Bt') S{R Q') At' = 0,
where A, B,G are the three determinants in
I, t"'. 3t", W I
Its — s"" t 
I ii" t" 
s"t'".
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65.] OF THE THIRD ORDER 195
SO that
^ = 9mi, £ = 3%, 6' = 3^,;
then solving the preceding equations for Q and for R — Q' in turn,
we find
3e=!^'i^ig)'/(M,.) )
and ' ' I,
2T(SQ) = 9°'6 °'°'t*°'  + sf)'f(».'.^)
Ill W[^ \wi/ ;
say. The latter equations may he regarded as the equivalent of
the two equations, which have been solved ; and therefore we may
expect that
/ / «' + f>'s + c't a"+b"3+c"t ^]^j(^ f ^j
r faf + b's + c't a" + 6"s +c"t \ r , ^ ,
\a + bs + ct a + bs + ct J \ ■ ■ "
the actual verification, which is comparatively simple, is left as an
esercisa Clearly these are generalisations of the property of the
Schwarzian derivative, represented by
[cs + d j
The two invariant functions / and J were first indicated* by
Painlev^ ; they subsequently were simpUfied to a form, which is the
equivalent of the above, by Boulangerf.
The invariance of the functions / and J, as indicated, exists
for lineolinear transformation of s and t. There is also an
invariance for any transformation of the independent variable z ;
for we easily find the equations
I(s, t, z)^I{s, t, Z)Z'+2{Z, z],
J(s, t, z) = J(s, t, Z) Z''~9I (s, t, Z) Z'Z" ~ 9 ^ (^. ^1
where Z is any function of z. Also
^I'(3,t,2
' Comptes Efndus, I. oiv (1887), p, 1830.
+ See his Thftse, Contribution h I'elude des ^q'uatiimB differentielies Un$a
it lumioginei intSgrabUs algSbriquement, [Paris, GautliierVillarfl, 1897).
13—2
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196 INVARIANTS AND [65.
and therefore
j(s, t, z)+^r{s, t. .) = [J(s, (, Z) + ^r{s. t, Z)\7i\
is an invariant for any change of the independent variable s.
Dropping a numerical constant, this is the function
which is the known Lagnerre invariant in the formal theory ; that
is*, if the equation
be transformed, by the relation
(sr
to the form
then
As the transformation
»f=(^.ii)(SJ'
'dZ\^
1' yii)
leaves the quotient of two integrals transformed only as by a
lineolinear substitution, it follows that the preceding function, say
L (s, t, s) = J is, t, z) + f 7' (s, t. z),
is unchanged by lineolinear transformations effected on s, t;
also, except as to a factor Z'', it is unchanged by transformation
effected on the independent variable. Now
30 that we have
" See a paper by the author, Phil. Tram. (1888), pp. 383, 390, Lagnerre's
invariant was first aanounoed in two notoe, Comptee Hendus, t. Lsixviii (1879),
pp. 116—119, 224—227.
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65.] FINITE GROUPS 197
which is the full expression of Laguerre's invariant in terms of
the derivatives of s and (,
66. The next stage is to associate these invariants with the
algebraic equations in two variables, which admit of one or other
of the finite groups. These groups have been obtained by Jordan*
and Valentinerf ; and references to other writers are given by
Boulangeri A method of using the results is outlined by Pain
lev^g as follows.
Let (s, t), 1^ (s, () denote two irreducible invariant functions
of a finite group of order JV; the functions are given by Klein for
the group of order 168, and by Eoiilanger (I.e.) for the group of
order 216. As these functions are invariable for each substitution
of the group, and as s, t are algebraic fimctions of s, it follows
that and i^ are rational functions of s, say
</,{.,() = * (4 f(s,t)=^^{^).
Conversely, taking * and ^ to be arbitrary rational functions of z,
these two equations give rise to N sets of simultaneous values
of s and t as algebraic functions of z ; and if any one set of
values be represented by <t, r, all the others are obtained on
transforming a and t by all the iV  1 substitutions of the group
other than the identical substitution. These two equations are
used to obtain the first four derivatives of s and ( with regard to s ;
and with these derivatives, the two invariants
I(s,t,^). J{s,t,z)
are constructed. The functions so formed involve derivatives of
<!> and ^ ; and the coefficients of these quantities are rational in
the derivatives of <^ (s, t) and i^ (s, t). As 7 and J are invariantive
for the group, the coeiEcients specified are rational functions of s
and t, which must be invariantive for the group and are therefore
rationally expressible in terms of ^ and t/t, that is, in terms of <&
• Crelk, t. Lxxxiv il878), pp. 89—215 ; AUi delta Jl. Accad. di Napoli, t. viii
(1879), No. II.
t Kj^b. Videmk. SeUk. Skr., 6 E., t. v (1889), pp. 61—235.
X In the Ttese, already cited on p. 19S, note.
% Comptei EendMS, t. civ (1887), pp. 1829—1832, ib, t. cv (1887], pp. 58—61.
II Math. Ann., t. xv (1879), pp. 265267.
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198 ALGEBRAIC [66.
and ^, Thus I(s, t, e) and J{s, t, z) would be expressed as
rational functions of s. Accordingly, taking
3Q = /(s,(,2),
R = hI'{s,t.z)i,J{s.t,z),
we have the differential equation
y'" + my' + Ry = 0.
The earlier investigations shewed that its integials are expressible
in terms of s, t, and their derivatives ; and we thus have a method
of constructing all the linear differential equations of the third
order which are integrable algebraically. There is a double
arbitrary element for each group, viz. the arbitrary forms of the
rational functions $ and 'P ; and there is a limited number of
groups,
67. While this outline is simple enough in general descrip
tion, the application to particular cases requires extremely elabo
rate calculations. These have beeti effected by Boulanger for the
group of order 216 ; they do not appear to have been yet effected
for any one of the other groups. As, however, the enumeration
of the finite groups in two quantities s aud ( is complete, the
subject offers an. interesting, if a laborious, field of investigation.
In the absence of the complete table of equations, for all the
finite groups and for two arbitrarily assumed functions ^ and "^j
it is not possible to use a method, analogous to that of § 62, to
determine whether a given equation of the third order is algebrai
cally integrable or not; it is not even possible to recognise to
which of the groups it would belong if it were algebraically
integrable. Indications of two general methods of procedure have
been given by Painlev4 and have been developed to some extent
by Boulanger; but the methods, while general in description,
suffer from the same kind of difficulty as the method indicated
for the construction of the equations, for the calculations are
exceedingly laborious. We have seen that, if two particular
values of s and (, say t aud t, are known, then an integral of
the differential equation is given by
J/ = (o'VtV')*.
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67.] INTEGRALS
Hence, if we take
so that the number of values, which u can acquire, is equal to M
or to a Hubmultiple of N, where JV is the order of the associated
group; let the number of values be n. Now if y ia algebraic,
every zero of y and every infinity of y are of a finite order, which
is commensurable in every instance ; and therefore all the infinities
of u are simple poles with commensurable residues. Substituting
for u in the ecjuation
y'" + ^Qy +Ry = 0,
we find
u" + Suu' + w^ + SQu + R=0,
a nonlinear equation of the second order satisfied by u. This
equation renders it possible to test the character of the poles and
the residues of u. If these are of the appropriate type, then the
equation is satisfied by a relation of the form
where A^, Ai, ,.., An are polynomials in s, and A^ is the product
of the factors corresponding to the poles of a. Then there is the
further test that this algebraic function u must be such that
is algebraic. Manifestly, the calculations will generally be too
elaborate to make the method eifective in practice.
Equations of tub Fourth Order.
68. As pointed out* by Painlev^ the processes just indicated
can formally be applied to linear equations of any order: but of
course, if any advance towards final conditions is to he made, it is
necessary to know all the finite lineolinear groups of transforma
tions in a number of variables less by one than the order of the
' Compls Eendua, i. cv (1887), p. 59.
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200 EQUATIONS OF THE FOURTH ORDER [68.
equation. Towards this enumeration of groups in three varia
bles, which are associated with the linear equation of the fourth
order, Jordan* has constructed a characteristic numerical equation
which, when completely resolved, would indicate the order and
the composition of each such group : but the resolution is exceed
ingly long and, owing to the number of cases that must be
considered, it has not been completed. In these circumstances,
no detailed results of a final critical character can be obtained for
an equation of the fourth order or of any higher order : the only
results obtainable are of a general character, and arise through the
association of groups in general with linear equations.
The equation of the fourth order, which may be written
w"" + 4pMi"' + 6}W!" + 4rw' + swi = 0,
can be transformed by
^g/p<i= ^ y
into
f" + eQy" + iUy' + 8^0.
We denote a system of four integrals by j(i, (/,, y,, y^, and we
introduce three quotients s, t, u, such that
then s, (, u are simultaneous solutions of three equations of the
fifth order in the derivatives. If a, r, v are a special set of
solutions, then
yi =
i
yi = yi<y, Vi = y^t, y^ = ^if ■
The complete primitive of the three equations is of the form
s _ _ _t u
a' + b'a + c't + rf^ ~ tt" i 6'V 4 c"t + d'V ~ a'" + h"'<T + d"r +
' Atti della B. Accad. di Napoli, t, viii (1870). No. 11, p. 25; instead of
dealing with lineoluiear traasformations in three vacia.bles, Jordan deals ivith
homogeneouB linear aubstitutioiiB in four variables.
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68.] WITH ALGEBBAIC INTEGRALS 201
There are three functions of the derivatives of s, t, u, with regard
to z, which are invariantive for substitutions such as the precodiug
relations expressing s, (, u, in terms of a, t, v; and they are
equal to
If the determinants
S ± {8"'t"u'\ % ± (s"t"u'), S ± {s''t"'u'), S + {s't'V)
be denoted by p, pi, p.^, p^ respectively, then
say ; if, in addition, the determinants
£ ± (s'Tu"), S ± (s'f'u')
be denoted by p^ and p^ respectively, then
say ; and if the determinant 2 ± (s''s"'s") be denoted by p^, then
Sso.= & ,„„„,„„„
«3;8e=f;i87»*+2W.
say. The three quantities I^ (s, t, u, z), /g (s, (, u, z), I^ (s, t, u, z)
are unchanged when lineolinear substitutions are effected on
s, t, u; and the combinations
/.+ 2/,l/,"A/,',
are also unchanged, except as to a power of Z', when e is replaced
by Z, any function of z.
The proofs of these various statements are left as exercises.
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202 algebraic integrals and [69.
Equations, having Algebraic Integrals, associated with
Homogeneous Forms.
69. It has already {§ 58) been stated that the discussion of
the equations, which have algebraic integrals, has been associated
with the theory of homogeneous forms : the association can be
seen to occur as follows.
Using the preceding notation of §§ 63 — 66 for the quantities
connected with any linear equation of the third order, we denote
by s and ( the quotients of any two by the third out of any three
linearly independent integrals of the equation
If, then, all the integrals of this equation are algebraic, both s
and t are algebraic functions of z ; they may therefore be
regarded as determined, in the most general case, by a couple of
distinct algebraic equations, say
/.(»,(, »).0, /,(s,i, «) = 0,
or by
9,{s.^) = 0, g,{t,z) = {).
Eliminating z between the pair of equations in whichever form
they are taken, we obtain a relation of the type
i^i,(s, = 0,
where Fg is a non homogeneous polynomial in s and t, because it
is the eliminant of two polynomials. Replacing s and i by jij ^ yi
and y^iyi respectively, and multiplying by the proper power of
3/1 to free the equation from fractions, we have
^ (yu y2, y^ = ^y,
where f is a homogeneous polynomial in its arguments or, in
other phrase, is a ternary form in 3/,, y,, y,.
Further, the above form of equation is obtained from
dht) , „ (iHv , „ dw , „
■j T + 3jo , T +3q:j + rw = 0,
by the transformation
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69.] HOMOGENEOUS FORMS 203
and therefore
that is,
F{vii, Wj, M'i) = 0,
on rejecting the i'actor e'"/?*', which occurs because F is a ternary
form (say) of order in. Hence it follows that when tlie integrals of
a liiiear equation of the third order are algebraic Junctions, a
homogeneous relation of finite order exists among any three linearly
independent integrals.
Moreover, when any other set of fundamental integrals F,, Y,,
Y, is taken, we know that
y, = a,Y, + a,Y., + a^Y,
y^=h,Y, + b,Y, + b,Y,
where the coefficients a, b, c are constants. The variables in the
homogeneous ternary form are therefore subject to linear trans
formation; and thus the theory of ternariants can be associated
with those homogeneous linear equations of the third order, which
have tbeii' integrals algebraic. The various cases will arise
according to the order of the form F; this order is always
greater than unity, because the integrals considered aie linearly
independent.
If, still further, we choose to combine the geometry of the
ternary form with the form in its association with the equation,
then the preceding algebraic relation ^ = is the equation of an
algebraic plane curve referred to homogeneous coordinates r the
curve is usually called the integral curve.
equation of the fourth
Wo
order
may
proceed similarly with a
n ec
dw
when all its integrals are algebraic. If we choose, we may trans
form it by the relation
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204 EQUATIONS OF THE
the quantity e^^^'^ must be aigebraic, because
where (7 is a non vanishing constant ; and the equation in y,
which is of the form
dz' dz^ as ■'
has all its integrals algebraic. Taking any four linearly inde
pendent solutions yi, y^, 1/3, y,, and writing
then as p, a, t are algebraic functions of z, they must be given
by three equations of the form
or of simpler equivalent forms, which are completely algebraic in
character. Eliminating z between the first and second, and also
between the first and third, and taking the eliminants in a form
free from irrational quantities if these occur, we have two
equations
F,{p,,,T)=0. ff.(p.^,T)=0,
two non homogeneous polynomials in p, a, r. Replacing these
quantities by their values in terms of ^1,^2, 1/3, 2/4, and multiplying
ea«h equation by the power of 1/,, appropriate to free it from
fractions, we find
where F and G are homogeneous polynomials in their arguments
or, in other phrase, are quaternary forms in 7,, y^, y^, ya As in
the case of the cubic, these equations imply the fiirther equations
F(Wj, Wa, Wj, W4) = 0I
so that, when the integrals of a homogeneous linear equation oj the
fourth order are algehraic Junctions, two homogeneous relations of
finite order earist among any four linearly independent integrals.
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69.] FOURTH ORDElt 205
Again, when the variables i/j, ys, y,, y^ are replaced by any
other set of fundamental integrals F,, Y^, Y^, Fj, the two sets of
variables are connected by homogeneous linear relations; and
thus the theory of quaternariants can be associated with those
homogeneous linear equations of the fourth order which have
their integrals algebraic. The various cases will arise according
to the orders of the forma F and G; these orders are always
greater than unity, because the integrals y,, y^, y^, y^ are linearly
independent.
We may also combine the geometry of quaternary forms with
the forms themselves as associated with the equation. In that case,
each of the equations F= 0, G = is the equation of a nonplanar
surface in three dimensions referred to homogeneous coordinates :
the two equations combined determine a skew curve, which ac
cordingly is the integral curve.
Similarly, in the case of equations of the fifth order, of which
all the integrals are algebraic, we have three homogeneous non
linear relations among any fundamental set of integrals ; and there
are corresponding associations with the theory of homogeneous
forms in five variables and the allied geometry. And so also for
linear equations of higher orders.
Note 1. There cannot be two homogeneous relations among a
set of three linearly independent integrals of an equation of the
third order: for they would determine a limited number of sets of
constant values for the ratios y, : y^: y^, contrary to the postulate
of linear independence.
Similarly, there cannot be three homogeneous relations among
a set of four linearly independent integrals of an equation of the
fourth order; for their existence would imply a corresponding
contradiction of the same postulate. And so for other equations
of higher ordeis.
It might however happen that, for an equation of the fourth
order, only a single homogeneous relation exists among four
linearly independent integrals; that, for an equation of the fifth
order, the number of homogeneous relations among a fundamental
set of integrals is less than three ; and so on. If the relations thus
given in each of the respective cases are the maximum number of
homogeneous relations that can exist, we can infer that not all
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206 BINARY [69.
the integrals of the respective equations are algebraic: and a
question arises as to the significance of the respective relations.
iVofe 2. The converse of the general argument must not be
assumed valid : that is to say, the existence of a homogeneous rela
tion between the members of a fundamental system of integrals
of an equation of the third order is not sufficient to ensure the
property that all the integrals are algebraic. Thus we know
that a number of transcendental functions of a variable can be
connected by algebraic relations : and such instances are not the
only possible exceptions.
70. The preceding method of a.ssociating the theory of forms
with linear equations does not apply directly when the equation
is of the second order : for a homogeneous relation between two
integrals would imply one or other of a limited number of con
stant values for the ratio of the integrals, which accordingly
could not be linearly independent. This deficiency, however, is
rendered relatively unimportant, because Klein's method explained
in §1 59^62 for the equation of the second order gives the
complete solution of the question propounded as to the cases
when ail its integrals are algebraic. The results there given
can be (and have been) obtained by processes directly connected
with the theory of binary forms. After the preceding exposition,
the analysis is mainly of formal interest, and adds little to
the knowledge of the solutions regarded as functions of the
independent variable.
It will be sufficiently illustrated* by one or two examples.
Ex. I. We tako the differential equation in the foi'm
and consider the value of a homogeneous polynomial function of two integrals
^1 aud y^i linearly independent of one another. Let this polynomial be of
order re, and write
* For fuller diBOUsaion and details, see Faaks, Crelle, t. Lixii (1376), pp. 97—
142, f6.,t. Lxxxv (1878), pp. 1—25; Briosclii, itfalft, ^nn., t. zi (1877), pp. 401— 411;
Forsyth, Quart. Journ., t. ixm (1889), pp. 45—78.
A memoir by Pepin, "Methode pour obtcnirlesintfigralesalgSbriques des Equations
dififeentialles lin^aires du seeond ordre," Row. Ace. P. d. N. L., I. isxiv (1883),
pp. 243 — 389, may slso be consnlted with advantage.
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70.] FORMS 207
saj, where s is the quotient ^,^^3. When substitutioa is made for i/j and
^2 ill terms of x, let the vahie of/ be <p (x), so that
Now \i R{yi,y^ = H{f) be the Hessian of/, and if H{u) ho the Hessian
of a, so that
^W"<'>(»S<»')(S)'}
We have also
*!
*■
■"&;
y,SJeon.i
say, so that
■£=^
Now
j,«*Wi
• *,
, c 1 <;»
^S <^
Differentiating, and aubatituting for the second derivative oiy^, we have
y^\dx J j/ dx u ds y^ d^ ds^
Multiply by n, and add the squaj^s of the aides of the preceding equation :
^n^21 L d^logu) \_ fdu\^\ _ I /d<p\\ d^{los<i>)
The coefficient of CVa"' on the lefthand side is
60 that
the Heaaian in terms of functions of x : let this be written
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208 EXAMPLES [70.
If now * (y^, y^) denote the oubiuo variant of/, so that
1 rofdH of a//)
then, proceeding in a similar way, we find
And so for other covariants.
As a special case*, let it be required to find tho value of 0, if when the
binary form is tho quadratic
a„ylH2a^3/lya + a32'2^
c^ {x) is a root of some rational function of x. In this instance,
a constant ; hetice i^ {:c) is either a rational function, or is the square root of
a rational function. The integration is immediate ; for
d$ Cdx
a^s^ + ^a^s + a^ if>ixy
The value of s is thus known : and the consequent values of y^ and y^ a
immediately given +.
.Ec. 2. Shew that, if the integrals of the equation
and is a root of some rational function of x, then <^* must be rational ; and
obtain tho relation between / and if) (x).
Ex, 3. The integrals of the equation
and (a;) is a root of some rational function of x ; shew that, unless ^ {x) is
actually rational, the quadrinvariant of the binary quartic must vanish. In
either case, find the relation between I and 1^ {x). (Brioschi.)
* Fuchs, CreiU, t. lxxsi (1876), p. 116.
t See my Treatise on Differential Equations, % 62.
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70.] OF BINARY FORMS
Ex. 4. Find the value of / in the equation
when, in the relatioii
connecting two integrals, the function ^ is supposed known.
Ex. f). Shew that, if two integrals of the equation
are connected by a relation
where A, B, C, D are cojistaiits, then
Assuming the condition satisfied, integrate the equation.
Ex. 6. Two integrals of the equation
are connected by a relation of the form
Ay^^^By^h,^ + Gy,y^^+Dyi^E=0,
where A , B, G, D, E are constants : prove that
d^Q .„dQ
z(P^Gl^q=Q.
Shew that the quantity on the lefthand side of this conditional equation ia
invaiiantive for change of the independent variable ; and hence, assuming
the condition satisfied, shew that the equation can be transformed so as to
become a particular case of Lamp's equation (Chap. ix). (Appell.)
Equations of the Third Order and Ternariants.
71. Returning now to the differential equation of the third
order in the form
and supposing that all its integrals are algebraic, we proceed to
consider the equation
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210 EQUATIONS OF THE THIRD ORDER [71.
where F is a homogeneous polynomial in any three linearly
independent integrals. For this purpose, it will be convenient
to have an equivalent simpler form of the equation which is given
by a known transformation*, viz, we have
a;+^»'
dt
■ 'iih^if
we take
I
the last of these relations may be replaced by the equation
The equation among any three integrals is
Consider the simplest case ; it arises when «. = 2, so that F is
then a quadratic polynomial involving six terms. Writing
a« = a,u, + «iMa + (laMa,
where a„ a^, a^ are umbral symbols, the equation can be symbolic
ally represented by
We have
where u' is du/dt, and so for u". Differentiating again, and
replacing u'" by — lu, we have
that is,
<*„'««" = 0,
on using the original equation. Similarly, on differentiating this
result,
 la^aa + Ou"^ = 0,
that is,
«„'■= = 0,
" See a paper by tlie author, Fbil. Tram., (1888), p. 441.
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71.] AND TERNARIANTS 211
Oil using the first derivative of the original equation. Differen
tiating once more, we have
la^au" = 0,
80 that either / = or a^Ou" = 0,
If / is not zero, then we must have
and therefore, by the second derivative of the original equation,
au^ = 0.
Hence, on the present liypothesis, we have
a^^ — 0, aaa„' — 0, a„^=0, auOu" = 0, au'aH" = 0, oi„"' = 0.
Now each of these equations is linear and homogeneous in the
six real coefficients that occur in a^^; eliminating these coeffi
cients, we obtain, as equal to zero, a determinant which is the
fourth power of
M2 . ■
and the latter ought therefore to vanish. But because w,, u^, Ms
are linearly independent, this determinant (being the determinant
of a fundamental system) dues not vanish — it is a nonzero
constant in the present case. Accordingly, the hypothesis that
/ ie not zero is invalid.
Hence i" = ; and therefore, on returning to the original
equation, we have
Writing
our original equation becomes
dz^ dz dz ^
Any three linearly independent integrals are connected by a
quadratic relation
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212 TEENABIANTS [71.
To obtain the integrals, we note that one value of m is a constant,
say unity ; thus
where
Thus three integrals of the original equation are ^i^, O^O^, 8^,
where fi and d^ are two linearly independent integrals of the
iatter equation of the second order.
It may be noted that three independent integrals of the
Mequation ai'e 1, (, f\ so that
dt ^ dt , dt
I'd,^ "'S'' I'di"'
and therefore
y^i  Vi'h = 0'
thus verifying the existence of the (quadratic relation obtained in
a canonical form,
Assuming known, we have
dz~ 6^'
so that
and thus three integrals of the original equation are
e. e.l%. ../f.
The comparison of these integrals with 6^, Oid^, 6j' is immediate ;
for it is a wellknown theorem that, if ^i is a solution of an
equation
g + pe.o,
then another solution, which is linearly independent of Sj , is given
by
Denoting this by 0^, the above three integrals are at once seen to
be ft", eA, «.'.
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71,] EXAMPLES 213
Ex. 1. Prove tha,t, if a be a solution of the equatioD
the primitive can be expressed in the form
i/ = Au + Binis.i){a I — J+(7«espf — a — J,
wbere A, B, C are arbitrary constants, and a ia a determiuate constant.
What 13 the primitive when a vanishes ? (Math. Trip. Part i, 1895.)
Ex. 2. Prove that, if three linearly independent integrals of the equation
be connected by a relation F{;y^, y^, 3'3) = 0, where ii™ is a homogeneouiS
polynomial of the third degree, then / muist satisfy the equation
(567'=  48/7") 7"'+5477"'« 1447'/"i'"+ IS^ . 7737'" +^ 24S7'3
 7 . ZQ^Prr + 84"77'S + ^^^^ /* = 0.
Ex. 3. Prove that, if both the fundamental invariants* of an equation of
the fourth order vanish, so that it can be taken in the form
y hlOPV +l^P'y +{'il' +97«)./ = 0,
then fDur 1 neirlj mdejiendent integrals are given by 8^, 6^6.^, 6i6.^\ 6^,
where S^ ml fl ire Imeiily indpi>endent intetjrali tf
Shew also that, if the relations
* These arise in the aame mannec as for the oubio. It the etiuation
be transformed by the relations
"4 + 40„p + Q,« = 0,
sS^^=0^
and the fundamental invariants are Os' ^»~^(ii'' ^^ "^ '
p. 210, note.
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214 EQUATIONS OF THE THIRD ORDER [71.
sulsist among four linearly independent integrals of aa equation of the
fourth, order, (so that the integral curve is a twisted cubic), tho equation
must he of the above form.
Ea:. 4. Construct the equation of the fourth order having fli^i, 5i0ai
6^1, ^2^3 for a set of linearly independent integrals, whore 6^ and fl^, p^ and
^2, are linearly independent int^rals of the respective equations
s+™". sa+e*"
Hence infer the form of a quartio equation when a single homogenec
quadratic relation subsists among a fundamental system of integrals.
Ex. 5. Shew that the equation
y"' + ry"+4^' + (6s' + 4r.s)y + 3(s"+!s')j=0
is satisfied by ;/ = ^, where 8 is an integral of
6"+sB=0;
and hence integrate the equation. (Fan
Es. 6. Shew that, if five linearly independent integrals of an equation
the fifth order are connected by the relations
[1 ^1. Vi, Vs, 3'* =0.
11 ^2. ys> ^'4. ^i I!
the equation can be taken in the form
SjaoSjiioiSft
.:;e('»£'')i(*S+«'£)»»^
and thence integrate the equation as far as posisible. (Fano.)
72. Consider now the more general case when three linearly
independent integrals of the equation
are connected hy an iiTesoluble relation
*■&.,</„ y.) = o,
where J*' is a homogeneous polynomial of order greater than two :
the question is as to the character of the integrals of the equation.
For the discussion, it is assumed that the differential equation
has its integrals regular and fi'ee from logarithms: it thus is of
Fuchsia n type.
Let K denote any non evanescent covariant of the quantic F;
such a covariant is the Hessian, which would vanish only if F
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72.] AND TEUNARIANTS 215
contained a linear factor. Let z describe any contour, which
encloses any one of the singularities, and return to its initial
value; the effect upon the fundamental system of integrals iji, y^,
yn is to change them into another fundamental system Fi, F,, F3,
the two systems being connected by relations
Y,= a,y, + ^,y, + rf,y„ (r=l, 2, 3).
The determinant of the coefficients a, /3, 7 (say A) is different
from zero in every such case ; in the present case, owing to the
absence of the term in A^ from the equation, we have {§ 14)
A = l,
by Poin care's theorem.
Now the preceding relations constitute a linear transformation
of the variables in the foregoing homogeneous forms ; hence if ^
be the index of K, and ^denote the same function of Fi, Fj, F;
as .ff" is of y,, j/g, j/j, we have
= K,
for fi is necessarily an integer. It thus appears that the value of
K is unaltered by the description of the contour.
This holds for each of the singularities, as well as for s = qo ;
hence K, when expressed as a function of z, is a uniform function.
To obtain the form of K in the vicinity of any singularity a, we
take account of the fact that the equation is of Fuchsian type :
hence in the vicinity we have, for any integral y,
(e — a)~py = hoiomorphic function oi z — a,
where ^ is a finite quantity. Now K is of finite order in the
variables 1/1, y,, y,; accordingly substituting for them, and remem
bering that ^ is a unifoi'ni function of 2, we have
{z — a)~''K = hoiomorphic function of « — tt,
where <r is an integer, positive or negative. This holds for estch of
the singularities, the number of which is limited when Q and R
are rational functions of 2 ; it holds also for z=ci:> . Hence K is
not merely a uniform function, but it is a rational function, oi z.
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216 APPLICATION OF [72.
It therefore follows that every eovariant of the quantic F is
a rational function of z, exceptions of course arising in the case
when the eovariant in an invariant, so that it is a mere constant.
Take then any two covariants, say the Hessian H, and any
other, say K : we have
where and ijf Eire rational functions of z. These are three
algebraical equations to determine y,, y^, y^ in terms of z; and
therefore the differential equation is integrable algebraically, a
theorem first announced* by Fuchs.
A case of exception arises, when the Hessian is a constant : the
quantic F is then of the second order so that the case has already
been discussed ; the integration of the original equation depends
upon the integrals of a linear equation of the second order.
As an illustration, consider the equation
wheii a fundamental set of integrals is coanected by a homogeneous cubic
relation. We assume that the equation is of Fuchsian type.
Talcing the cubic in the canonical form, we have
I being a constant. The Hessian is a rational function, say 1^(1+8?^); so
ir=f (y.'+y,' +y,=)  (1 + 2z=) y,y,j<3 = <j. { 1 + 8;«),
and therefore
Taking the other symmetric covariantt of the cubic, which also ia a rational
function, we have
and * is equal to a rational function ; so that, ta.kiiig account of the above
value of ^I'+ya^+^s', we can write
Thus jijS, ^^^, yj' arc the roots of
' Acta Math., t. I (1882), p, 830.
t Cayley, Coll. Math. Papers, t. si, p. 345,
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72.] COVABUNTS 217
an irreducible cubic. So far aa the coefficients are concerned, they are known
to be rational functions of z ; the denominator of each such function is known,
because its faotora arise through the aingularitiee of the equation and the
multiplicity of any factor can be determined through the associated iadiciaJ
equation ; and the degree of the numerator has an upper limit, determined
by the behaviour of the integrals for large values of z. Heace and ■^ can
be regarded aa known, save aa to a polynomial numerator in each case.
We have
Tl"' = Afi^ + Bir, + a^ ]
the last three being obtained, after diftereatiation, by repeated use of the
cubic equation for ij, and the quantities A, B, G, ... being functions of </i, i/'
and their derivatives Now writing y = i^ in the differential equation, we
find
When the above values are substituted and the result is reduced by means of
the cubic equation, so that no power of ij higher than the second occurs, we
have an equation of the form
where Y,, Yj, Y, involve 0, ijr and their derivatives, and are linear in Q, R.
As the cubic is irreducible, so that this equation holds for each root, we have
Yi = 0, Ya^O, Y3 = 0,
three equations to determine and ■^. There consequently exists a relation
among the remaining quantities, viz. Q and R : and this must be equivalent
to the condition (§ 71, Ex. 2), which must be satisfied in order that the
equation .^=0 may exist.
Similar results hold for the cubic equation, when the homo
geneous relation between the integrals is of order greater than
three ; and corresponding results hold for linear differential
equations of higher orders. In fact, if a general homogeneous
relation of finite order higher than the second subsists among a
fundamental si/stem of integrals of a linear differential equation of
order n, then the equation is integrable algebraically: the proof
follows the lines of the preceding proof exactly.
This range of investigations will not, however, be pursued
further, as it becomes mainly formal in character, depending upon
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218 APPLICATION OF COVARIANTS [72.
the theory of covariants and upon the application of the theory of
groups to linear differential equations. An excellent account of
what has been achieved, together with many references, is given
in a memoir* by Pano who has made many contributions to tho
subject ; a memoirf by Brioschi contains some investigations con
nected with ternariants; and other detailed references are given
in Schlesinger's treatise^, which contains ati ample discussion of
the subject.
' Math. Ann., t. Liri (1900), pp. 493—590.
+ Ann. di Mat., 2" Ser., t. xm (1885), pp. 1~21.
J Tkeorie der linearen DiffeTentiatgUiekimgen, ii, 1 (1897), pp. viii — si. The
diBCUssioti is to lie found in ohupters 2 — 6 of the tenth seotlon of the treatise.
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CHAPTER VI.
Equations having only some of their Integrals regular
NEAR A Singularity.
73. It has been seen that, if all the integrals of an equation
are to be regular in the vicinity of each singularity, the coefficients
in the equation must be rational functions of z of appropriate
form and degree.
It may, however, happen that the coefficients are rational
functions of s but are not of the appropriate form and degree :
in that case, it is not the fact that all the integrals are regular,
and it may even be the fact that none of the integrals are regular.
This deviation from regularity need not occur at each singularity
of the equation : a fundamental system may be entirely regular in
the vicinity of one (or more than one) of the singularities, and
may not possess its entirely regular character in the vicinity of
some other. The conditions necessary and sufficient to secure
that all the integrals are regular in the vicinity of a singularity a
have already (Ch. Hi) been obtained. If these conditions are not
satisfied, then the composition of the fundamental system in the
vicinity of the singularity a is no longer of an entirely regular
character; we desii'e to know the deviations from regularity.
It may also happen that not all the coefficients are rational
functions of z; in that case, if uniform, they are transcendental
functions and possess at least one essential singularity, say c.
Further, owing either to a possibly excessive degree of the
numerator in a rational meromorpbie coefficient or to a possibility
that z—<x: is an essential singularity of some one or more of the
coefficients, it can happen that the conditions for regularity of
integrals near a^oc are not satisfied. The fundamental system
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220 EQUATIONS HAVING [73.
is then not entirely regular near c or for large values of \z\, in
the respective cases indicated, and it may even be devoid of any
regular element; the same question as to its composition arises
as in the corresponding hypothesis for the singularity a.
Accordingly, for our present purpose we assume that the
coetBcients in the differential equation are everywhere uniform:
that (unless as otherwise stated) they may have any number of
poles, and that they may have one or more essential singularities.
When tt is a pole of one (or more than one) of the coefficients,
and is not an essential singularity of any of them, we have one
of the cases just indicated; when qo is a pole of coefficients,
not being an essential singularity of any one of them, we have
another. We write
1
in these respective cases ; and then our differential equation takes
the form
d^w d^'w d'^'^w dvj
where the point a; = is a pole of some (and it may be of all) the
coefficients. If a!I the integrals were regular in the vicinity of
ic = 0, then x'^p^ for r = 1, 2, ..., m would be a uniform function of
X that does not become infinite when x=0. As some of the
integrals are to be not regular in the vicinity of x = Q, the
multiplicity of the origin as a pole of p, must be greater than r,
for some value or values of r. Let
p^=a:'^'Pr{s>), (r=l, ..., m),
where m, is a positive integer (which may be zero for particular
coefficients), and Pr (*') is a uniform function of a: which does not
become infinite when a:=0: also it will be assumed that, unless
pr vanishes identically, ■=r, has been chosen so that Pr{0) does not
vanish, so that ct, measures the multiplicity of the pole of p, at
the origin. Then one or more than one of the quantities
w,.r (r = l,...,m)
is a positive integer greater than zero.
As in § 23, let
r 1 ,,....
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73.] ONLY SOME INTEGRALS REGULAR 221
and suppose that
■^lA + 0Ali + 0A5i>+ ... + 0it*~' + 0o£*
is ao integral of the equation, regular ill the vicinity of a; = and
belonging to an exponent fj. ; then it is known (§§ 25 — 28) that
<^„ is a regular integral also belonging to the exponent fs,, so that
where <!>„ is a uniform function of x which does not vanish when
x = 0. As this expression, when substituted for w, should make
the equation satisfied identically, the aggregate coefHcient of the
lowest power of x must vanish (as, of course, must all the other
aggregate coefficients). The lowest power of ic in the respective
terms has for its index
/j.~m, ij. — ^i—{m — i), fi. — 'a^ — {'in — 2), ..., /j. — ia^^i — 1, /j. — 'nr^:
and for any other integral, belonging to an exponent <7, the
corresponding numbers would be
a — m., (ri!Ti — (m ~1), 17 ~ t^s — (m — 2), ..., <r — OTm_i— 1, <r — st™.
Let
w. + (ms)=rj„ (s = 0, 1, ...,'m),
and consider the set of integers
n„, n., .... n^.
Of these, let the greatest be chosen. It may occur several times
in the set ; when this is the case, let the first occurrence be at
n„, as we pass in the order of increasing subscripts, so that
n^<n„ , for r = 0, 1, ...,n\,
n„^n^., r = 0, 1, ....mn.
Then n is called* the characteristic index of the equation : when
K = 0, all the integrals are regular.
The lowest power of x after substitution of the expression for
the regular integral has (U. — n„ for its index ; it arises through
p„ — y — and later terms in the differential equation ; as the
coefficient of this lowest power must vanish, the exponent fi must
* Thome, Cielle, t. lksv (1873), p. 267.
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222 CHARACTERISTIC INDEX AND INDICIAL EQUATION [73.
satisfy an algebraic equation of degree m — n. Similarly for an
exponent o to which any other regular ititegi'al belongs ; it also is
a root of the same algebraic equation; and each such exponent
satisfies that same algebraic equation of degree m — n, which
accordingly ia called the indicial equation. But it must not be
assumed (and, in fact, it is not necessarily the ease when n > 0)
that the number of regular integrals is equal to the degree of the
indicial equation. It is clear that, in all cases where n.>0, the
degree of the indicial equation is less than in.
71. Suppose now that the given differential equation of order
■m has a number s of regular integrals, which are linearly inde
pendent of one another, where s <m: (the case s = m has already
been discussed) : and that there do not exist more than s linearly
independent integials. After the earlier discussion of fundamental
systems, it is clear that any regular integral of the equation is
expressible as a homogeneous linear combination of the s integrals,
with constant coefficients ; also that, if every regular integral of
the equation is expressible as such a combination of s (and not
fewer than s) such integrals, the number of regular integrals
linearly independent of one another is s.
Further, a linear relation among the integrals of the equation,
involving a number of regular integrals and only a single one that
is not of the regular type, cannot exist ; for the single nonregular
integral would involve an unlimited number of negative powers of
w, while each of the others occurring in the linear relation involves
only a limited number of such negative powers.
A linear relation might exist among the integrals of the
equation, involving a number of regular integrals and two Integrals
that are not of the regular type. We then regard the relation as
shewing that the deviation from regularity is the same for the
two integrals : and in constituting the fundamental system for the
equation, we could use the relation as enabling us to reject one
of the nonregular integrals, because it is linearly expressible in
terms of integrals already retained. So also for a linear relation
with constant coefficients between regular integrals and more than
two integrals of a nonregular type.
Again, suppose that our differential equation of order m has
an aggregate of n integrals, regular in the vicinity of ic = and
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74.] THEORY OF BEDUOIBILITY 223
linearly iniiependent of one another; and let it be formed of sub
groups of integrals of the type
for'K — 0,l,2,...,ic, where
fA.■v^^,lZ'■' + ^^l,,i^
Then, after ^ 25 — 28, we know that these n linearly independent
integrals constitute a fundamental system for a linear differential
equation of order n, the coefficients of which are functions of ic,
uniform in the vicinity of a^ = ; let it bo
r^^^
'^'d^'^
+ i'j(^ =0
Now this equation, being of order n, cannot have more than n
linearly independent integrals i and its fundamental system in the
vicinity of gs — O is composed of the n regular integrals of the
original equation. Hence, by § 31, we must have
r^ = a>''B^(w), {fi = l, 2, ..., n),
where Il^{ic) is a holoraorphic function of x in the vicinity of
ic = 0, such that R^(0) is not infinite. Accordingly, the aggregate
of the n linearly independent regular integrals of the original
equation are the n integrals in a fundamental system, of a linear
equation of order n of the foregoing type.
Eeducibility Of Equationk.
75. If therefore some (but not all) of the integrals of the
given equation of order m are of the regular type, it has integrals
in common with an equation of lower order. On the analogy of
rational algebraic equations, which possess roots satisfying an
algebraic equation of the same rational fonn and of lower
degree, the differential equation is said to be reducible.
Consider two equations
) d™ '■y
*(!/)«.
*Sj
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224 BEDUOIBILITY OF [75.
■where m>n; and take an expression
where the coefficients fio, .B,, ..., Ri are at our disposal, and
Let these disposable coefficients bo chosen, so as to make the order
of the equation
JJ(y)iliT(!,)! =
as low as possible. By taking the ^ + 1 relations
P. = B.Q,.
p,=R,a+B.(ia'+a),
P, = ftft +B,{(ii) q; + Q,] + B. lii (i  1) Q." + iq; + a),
i'iae.+Bi.(a'+ft)+Ki.(a"+2Q.'+Q.)+...,
which determine R^, ■..,Ri, we can secure that the terms involving
derivatives of y of order higher than n — \ disappear. Accordingly,
writing
where S^, S,, ..., Sj are determinate quantities and
we have
where K is of order less than if. Moreover, if P„, ..., P^,
Qo, ■■■, Qn are uniform functions of x, having x = either an
ordinary point or only a pole, the same holds of the coefficients R
and the coefficients S ; so that L and K are of the same generic
character as M and N.
From this result several conclusions can be drawn,
I. Any integral, common to the equations Jf = 0, N = 0, is an
integral of the equation K = 0. If, therefore, every integral of
JV = is also an integral of M=0, it follows that K=0 must
possess n linearly independent integrals ; as its order is less than
n, the equation is evanescent, and we then have
«{s).ifiV(S,)).
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75.] AN EQUATION 225
II. Any integral, common to the equations JV=0, K~0, is
an integral of the equation Jf = ; and therefore, in connection
with the first part of the preceding result, the integrals common
to M^O, N = constitute the integrals common to i^'" = 0, K=0.
The process of obtaining the integrals (if any), common to
two given equatioDS ilf =0 and i\r = 0, can thus be made a kind of
generalisation of the process of obtaining the greatest common
measure of two given poiynomiais. Proceeding as above, we have
M =iiV +K \
K = L,K, +K,r
where K^, K.2, ..., Kg are of successively decreasing orders. Then
unless an evanescent quantity K of nonzero order is reached,
sooner or later a quantity K is reached which is of order zero,
that is, contains no derivative.
In the former case, let ^,+1 be evanescent ; then the integrals
of the equation Kt= constitute the aggregate of integrals common
to M = 0,N=0.
In the latter case, let Kg be the quantity of order zero ; then
the integrals common to M=0, ^ — are integrals of
lf, = y/W0.
Now f(z) is not zero, for otherwise ^s would be evanescent ; and
therefore we have
y = 0,
the trivial solution common to all homogeneous linear equations.
We then say that Jf = 0, A'" = have no common integral.
III. An equation having regular integrals is reducible. For
one such integral exists in the form
y— :•/(").
where ^ is finite, andy(fl;) is holomorphic in the vicinity of «= 0,
while /(O) ia not zero. We have
ydw X f{x)
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226 CHARACTERISTIC FUNCTION [75,
where R(x) is a holomorphic function in t.he vicinity of w = 0,
such that R{0) is not zero. Thus the given differential C(^iiation
has an integral satisfying the equation
that is, it has an integral common with an equation, which is of
the first order and is of the same form as itself: in other words,
the equation is reducible.
But it is not to be inferred that such equations are the only
reducible equations.
IV. If an equation M = has p (and not more than p)
linearly independent regular integrals, it can be expressed in
the form
where N is of order p, and L is of order m — p.
For the p regular integrals are known (§§ 25—28, 74) to
satisfy an equation of the form
of order p. Every integral of iV"=0 is an integral of M=0;
whence, by I., the result follows.
76. We proceed to utilise the last result in order to obtain
some conclusions as regards the regular integrals (if any) of a
given equation, say,
,^ , , d'"w! d'^^'w dw
The result of substituting x^ for w in P (w), where p is a constant
quantity, is
this is called* the characteristic /unction of the equation 7' = or
of the operator P. We have
... +p,^,^+Pv,;
* Frobeaius, Crelle, t. i.ssx (lS7a), p. 318,
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76.] INDICIAL EQUATION 227
when the righthand side is expanded in ascending powers of x, it
contains (owing to the form of the coefficients p) only a limited
number of powers with negative indices. The highest powers of
oT^, arising out of the m + 1 terms in i>;~''P(W), have exponents
m, OT,+mJ, OT2 + m2, ..., ct™_, + 1, vr^,
Ho, n., ..., n^.
Let n be the characteristic index of the equation, so that n„ is
the greatest integer in the set : if several of the quantities 11 be
equal to this greatest integer, then n^ is the first that occurs as
we proceed through the set from left to right. Denoting the
value of n„ by g, let
''^"ffl^"?'* (*')"?'•' ^'" ~ ^' "' ■■■' ^)'
so that 5„ (0) is not zero, and no one of the quantities q^ (0) is
infinite. Then
«.P(i')_^.G(p.«;),
where (? is a polynomial in p and is hoiomorphic in a: in the
vicinity of a; = 0. Moreover, expanding G(p, x) in ascending
powers of x, we have
Gip,^)=9,{p)+a:grip) + ...,
where each of the coefficients ^ is a polynomial in p, of degree not
higher than m; the degree of ffoip) is m — n, and the degree of
gg^mip) is ™ Also, g^ip) is the quantity called (§ 39) the indicial
function ; the equation
ft(f>) =
is called the indicial equation.
Now take
N(w)^a:ffP(w)
d^w ^ , d'""''!!! dw
 1^ £~ + *' £?T + ■ ■ ■ + «'^ S + *'"•
where q„ = xs~'^ ; the equation P — can manifestly be replaced
by the equivalent
J{»)0,
which is taken to be the normal form for the present purpose.
We have
arPN{af) = G{p,x) = g,{p)\xg,{.p) + ,
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228 NORMAL FORM OF EQUATION [76.
which thus contains only positive powers of x when the equation
is in its normal form, and which has the indicial function for the
term independent of x.
We have seen that, if P {w) = possess regular integrals, it
is a reducible equation : and the operator P can then be repre
sented as a product of operators. Consider, more generally in the
first instance, two operators A and B, each in its normal form ; and
let G, also an operator, denote AB. Further, let the characteristic
functions of A, B, G, respectively be
A (^) = a^^fia:, /,) = «^ 2 /. (p) ^^ = 2 /, (p) a:^+
B{a^) = a^g(x,p)=^x<' 2 g^ip)x'^= 2 g^{p)x'^+>
0(a:'') = xi'h{x,p)^af 2 h^(p)x''== 2 k^(p}x^+<,
where the summations in f(x, p) and g{x, p) include no negative
powers of fl^, because A and B are in their normal forms. Now, as
C= AB, we have
G{x') = AB{af)
^A[%j,{p)ar+']
= %^l^gAp)/>.(^+p)x^+'^+^
and therefore
S 4, (p) «'  ^X ^S J7, (p)/, (M + p) «="'.
As X and p. are incapable of negative values, there are no negative
values for a ; and therefore G is in a normal farm. Also
fh(p) = gi,{p)f<,{p),
80 that the indicial function of G is the product of the indicial
functions of its component operators : and
h,(p)= S^y„ (/>)/„_„ (/i + p).
Further, if C be known to possess a component factor B which,
when operated upon by A, produces C, then A can be obtained.
For, take B and C in their normal forms : the equation
S h,(p)i^~ t Xsr,(p)A(^ + p)a^+.
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76.] CHAEACTERISTIC INDEX 229
then holds. The values of X are clearly 0, 1, ..., so that A is then
in its normal form ; and the successive quantities fy are given by
the equation
for o = 0, 1, ..., p, the values obtained being polynomials in p,
because G is known to be composite of A and B.
Of course, this merely gives the characteristic function of the
operator ; but the characteristic function uniquely determines the
operator. For let /(ic, p) be a function, which is a polynomial
in p, and the coefficients of which are functions of x: and let
the degree of the' polynomial be m. Then we have*
where, taking finite differences in tlie form
4/(«.p) =/('»,(' + !)/(«,/>),
we have
<.!«. = 14/(«,P)1....
Tlius
which is the characteristic function of the operator
da?'* "* ' dai"'
. + v^x J + u„:
the operator is determined by the characteristic function.
Characteristic Index, and Number of Regular Integrals.
77. Now let the equation of order m, taken in its normal
form, be
„, , d^""w ,d"'hjj dw „
i\r (^) . ^^ _ + 5,.. ^^^ + . .. + 5™,«^ ^ + 3™«. = ;
and suppose that it possesses s (and not more than s) regular
integrals, linearly independent of one another. These s integrals
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230 CHARACTERISTIC INDEX AND [77.
are a fundamental ayatem of an equation, of order s and of Fuchs
ian type ; when this equation is taken in its normal form, let it be
^ ' d^ oaf' da>
where cr,, ctsi ■■. <^a ^^^ holomorphic functions of x in the vicinity
of x = 0. As all the integrals of jS' = are possessed by iV=0,
there exists a differential operator T of order m — s, such that
because N and S are in their normal forms, T also is in its normal
form, so that we can take
(M'"~' ax'" ' ' ax
where t,, T3, ..., Tms are holomorphic functions of ic in the vicinity
offl!=0. If then
Tixfy^xPBix.p),
the indicial function of 2' is the coefficient of x" in $ (x, p), which
is a polynomial in p and coutains no negative powers of x. This
coefficient may be independent of p ; in that case, the character
istic index of 2" is m — 5. Or it may be a polynomial in p, say of
degree k in p, where k^0\ the characteristic index of T then is
i N' = TS, the indicial function of N is the product of
the indicial functions of T and S; so that the ifidicial /unction of
S, which gives all the regtdar integrals of N, is a factor of the
indicial function of the original equation. The degree of the
indicial function of 8 is equal to s, because S = is an equation of
order s of Fuchsian type ; the degree of the indicial function of N
is m— w, where n is the characteristic index of ^=0. Hence
so that (assuming for the moment that k may be either zero or
greater than zero) an upper limit for the number of regular
integrals which an equation can possess is given by
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77.} THE NUMBER OF REGULAR I^TEGRALS 231
where m is the order of the equation, and n is its characteristic
index (supposed to be greater than zero). It is known that, when
n=0, the number of regular integrals is equal to m.
Corollary I. An equation, whose indicial function is a
constant, so that its indicial equation has no roots, has no regular
integrals; for its characteristic index is equal to its order. But
such equations are noC the only equations devoid of regular
integrals.
CoROLLABY II. When k is equal to zero, then s is equal to
m — n, so that the number of regular integrals of the equation
is actually equal to the degree of the indicial function. The
necessary and sufficient condition for this result is that the
equation, which is reducible, must be capable of expression in
the form
where the indicial function of ?■ is a constant, and tho degree of
the indicial function of S is equal to the order of S.
This result, which is of the nature of a descriptive condition,
appears to have been first given in this form by Floquet*. Other
forms, of a similar kind, had been given earlier by Thom^f and by
FrobeniusJ (see  83, post).
Note. On the basis of the preceding analysis, it is easy to
frame an independent verification that the characteristic index
is not greater than m — s. For in the operator T, the quantity
Tm^s^ic does not vanish when a: = ; and all the quantities r>, , such
that
\<m~ s — k,
do vanish when a: = 0. Hence, when we take iV as expressed in
the form
the coefficient of
is the first (in the succession from left to right) in which Tm„(_i
occurs; it also contains q^, t,, .... Tmssi, all of them occurring
* Ana. de I'^o. Nona. Sup., 2° S6r., t, vui (1879), Buppl, pp. 63, 64.
t Crelle, t. lxxvi (1S73), p. 286,
t GrelU, t. isns (1875), pp. 331, 832.
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232 NUMBER OF REGULAB INTEGRALS [77.
linearly. When a;=0, all of these except t™„j_j: vaoish, and
Tntn does not vanish ; and therefore qmai does not vanish when
a!=0. In the coefficient of
of , — ,
where iJ.>s\k, the quantities i/o, ti, ..., t^_„ occur linearly: each
of these vanishes when ic = 0, and therefore jm» does vanish when
ic = 0. As this holds for all values of /i, it follows that (jmsk is
the first of the quantities q which does not vanish when ic=0;
hence the characteristic index of N is in — b — k, that is, it is
^m—s, where s is the number of regular integrals possessed by
the equation iV = 0.
E<c. 1. If w=Mii be an integral, regular and free from l<^arithm8, of an
equation P=0, which is of order m and haa i regular integrak, and if a new
dependent variable u be given by
ahew that u satisfies an equation Q = 0, which is of order m  1 and has s — 1
regular integrals ; and obtaiQ the relation between the characteriatic index of
P=0 and that of §=0. (Thomd.)
Ex. 2. The equation
s integrals regular in the vicinity of  = and linearly independent of
th d=0 lltjJ P hwthtt pl(t
1 1 t ) f ea h f th m g uefii t p (Tl m )
b ti^ ly as gn d bj t t th d t th t =0 pi
d yptp thttl m wffi i p be d t n ed
a. t perm tth qt tp ss — bt lyaasged gila,
t gr 1 h ealj d p d t f th (rhm^)
^4P thtth dt ydflitt tht
luftt \=0 f i d 1 vmg d If t i d f,
yhllh yilealydpdt^l t 1 thtV
lllbepltfthfm QMD wheth d If i la i Q M D
are of degrees 5, 0, myh respectively, and D is of order myS. Is
there any limitation upon the order of Mf (Cayley.)
Ex. 5. Shew that an equation QD=0 has at least as many regular
integrals as Z> = 0, and not more than Q = and 5 = together ; and that, if
all the integrals of D=0 are regular, then QI>=0 lias as many regular
integrals as §=0 and i)=0 together.
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71.] AKD DEGREE OF INDICIAL FUNCTION 233
Hence (or otherwise) shew that, if an equation F=0 has all its integrals
regular, then F can be resolved into a product of operators, each of the first
order and such that, equated to zero, it has a regular integral. la this
resolution unique? (Frobenius.)
78. In the two extreme cases, first, where the degree of the
indicial function is equal to the order of the equation, and second,
where its degree is zero, the number of regular integrals is equal
to that degree. The preceding proposition shews that, in the
intermediate cases, the degree merely gives an upper limit for the
number of regular integrals. It is natural to enquire whether
the number can fall below that upper limit.
As a matter of fact, it is possible* to construct equations, the
number of whose regular integrals is less than the degree of the
indicial function. Taking only the simplest ease leading to equa
tions of the second order, consider the two equations
U = ^ + ky + h = 0, F=J + % = 0,
da! " dx "
of the first order ; and form the equation
ax ax
which manifestly is of the second order, say
d'y dy
If we can arrange so that ic = is a pole of p of order n, where
n ^ 2, then « = in general will be a pole of q of order ;i + 1 ; and
the indicial function will then be of the first degree.
Consider now the equation of the second order. Since
0" V^h,
it can be written
which is satisfied by
i — 
dx
where A is any ai'bitrary constant.
V (1872), pp. 311—313.
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234 THOMfi's THEOHEM [78.
Let Y be an integral of the equation of the second order. It
may be an integral of F = ; if it is not, then, when we take
that is, 1/, is an integral of U= 0. Thus any integral of the
equation of the second order either is an integral of 1^= or is a
constant multiple of an integral of ^7=0. If, then, U=0 and
V=0 are such that they possess no regular integral, the differ
ential equation of the second order can possess no regular integral ;
at the same time, its indicial function is of the first degree.
The equation V"= will not have a regular integral, if a; = is a
pole of k of order greater than unity ; and the equation U=0 will
then not have a regular integral, if /i is a rational function of x.
Ex. 1. The aggregate of conditions can be aatiisfled aimnltaneously in
many ways. For instance, take
's+i. «
Z+^x
si;
The differential equation of the second order is
d^y 1_ c^_3 + 2^_,
dx' ^ dx
its indicial equation is of tlie first degree, and it has no regiilar integrals : or
the number of its regular integrals is less than the degree of its indicial
equation.
The conclusion can otherwise be verified ; for it is easy to obtain two
linearly independent integrals in the form
17^
no linear combination of which gives rise to a regular integraL
Ex. 2. Shew that the equation
has no regular integrals : and verify the result by obtaining the integrals of
the equation, (ThomiS, Floquet.)
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79.] DETEBMINATIOS OF THE REGULAR INTEGRALS
Determination of such Regular Integrals as exist.
79. When the degree of the indicial function of an equation
of order m is less than m, no precise information is given as to the
number of regular integrals possessed by the equation. The further
conditions, sufficient to determine whether a regular integral
should or should not be associated with any root of the indicial
equation, can be obtained in a form, which is mainly descriptive
for the equation of general order and can be rendered completely
explicit for any particular given equation.
Let the equation be
A {w) = go^'" ^^ + q,x"^' ^^^ + . . . + <;™w = 0,
of characteristic index n. Let E{0)he the indicial function
, and
let a be one of its zeros, so that
Then, if a regular integral is to be associated with a, it n)ust be of
the form
w = a^(c„+ c,« + Csa;^+ ... + CpX^ + ...).
This expression, when substituted in the equation, must satisfy it
identically, so that, after substitution, the coefficient of «"+*■ must
vanish for every value of p : and therefore
where the number of terms in this difference relation depends
upon the actual forms of g,,, q,, ..., 5™. Of the coefficients /„/i,
.■■,fr, the first is
which is of degree m — n in p; of the remainder, one at least, viz.
fgm, is of degree m in p, where g has the same significance as in
§76.
The successive use of this difference relation, together with the
equations for the earlier coefficients, the first of which is
leads to the values of all the quantities (;^hc„, for the successive
values of fj. ; and thus a formal expression for it is obtained that
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23C DETERMINATION OF THE [79.
satisfies the equation. If, however, the expression is an infinite
series, it haa no functional significance when it diverges : that this
frequently, even generally, is the case, may be inferred as follows.
For if c^+i i c^, with indefinite increase of (i, tends to a limit that
is not infinite, so also would C,iaic„+i, C/^^^c,^^, and so on;
and therefore
for finite values of o, also would tend to a limit that is not infinite.
Now a number of the quantities
7.W'
for various values of 6, undoubtedly tend to zero as jj, increases
indefinitely ; some of them may have a finite limit ; but one at
least is infinite, viz.
/.w ■
because the numerator is of degree n higher than the denominator,
both of them being polynomials in /i. Consequently, the ex
pression
acquires an infinite value as i* increases without limit. The
difference relation requires the value of the expression to be
always  1, so that the hypothesis leading to the wrong inference
must be untenable. Therefore c,^ii c^, with indefinite increase
of /i, does not tend to a limit that is finite, and therefore the
series diverges*. There is then no regular integral to be asso
ciated with the root a.
* It is not inconceivable that, for special values of in and of n, and for special
forms of the eoefEoients q, as well as for a speoial value of the limit c„4.j4Cji, the
infinite parts of the expression
,J,/7W "•,
might (lisappear, and the espression itself be eiiual to 1. In that case, the
series would oonvei^e : and an exception to the general theorem would occur. But
it is dear that such an eioeption is of a yery special character: it will be left
without further attempt to state the conditions explicitly.
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79,] REGULAR INTEGRAIS THAT EXIST 237
As the series thus generally diverges when it contains an un
limited number of terms, the regular integral is thus generally
illusory. The only alternative is that the series should contain
a limited number of terms : and then the regular integral would
certainly exist. Accordingly, let it be supposed that the series
contains k + 1 terms, so that
Ci Ci Ct
are quantities known from the differencerelation, and that
Ck+i, c*+2. ■■■ ad inf.
all vanish. If we secure that cj+i, ct+i, ..., Cn+r all vanish, then
every succeeding coefficient must vanish in virtue of the diffei'ence
relation ; and these t relations will then secure the existence of a
regular integral to be associated with the exponent cr. Taking
p — k, k~l, ..., k — r + 1 in succession, we find the t necessary
conditions to be
/„ (k) Ci = 0, that is, /„ (k) = 0,
and generally
for values r= 1, 2, ..., t 1. The first of these is
/S(a + k) = 0.
so that the indicial equation, which possesses a root er, must
possess also a root <T + k, where k is a positive integer. (In the
special instance, when k = 0, no condition is thus imposed : in the
general instance, when & is a positive integer greater than zero, it
is easy to verify that E{a + k) is the indicial function for ic = x .)
When the aggregate of conditions, which will not be examined
in further detail, is satisfied in connection with a root of the
indicial equation, a regular integral exists, belonging to that root
as its exponent; and there are as many regular integrals, thus
determined, as there are sets of conditions satisfied for each root
of the indicial equation.
Explicit expressions for the various coeOicients c can be derived,
when the explicit forms of the quantities q are known : but the
general results involve merely laborious calculation, and would
hardly be used in any particular case. The results are therefore.
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238 MODE OF OBTAINING [79.
as already remarked, mainly descriptive : and so, in any particular
case, it remains chiefly a matter for experimental trial {to be
completed) whether a regular integral ia necessarily i
with a root of the indicial equation.
For this purpose, and also for the purpose of c
regular integrals associated with a multiple root of the indicial
equation, a convenient plan is to adopt the process given by
Frobenius (Chap. Ill) when all the integrals are regular. We
substitute an expression
W = CoClf It CiiC^+'+ ... +c^af^i^+ ...
in the equation
of characteristic index n. After the substitution, the ttrst term is
where E{p') is the indicial function, of degree in — n; and we make
all the succeeding terms vanish, by choosing the relations among
the constants c appropriate for the purpose. We thus have
N{w) = cE{p)a^;
and the relations among the constants c are of the form
where the constants ix^^^i' ■■■> '^co ^^'"^ polynomials in /i and,
when this relation is the general difference relation between the
coeiBcients c, one at least of these polynomials a^r is of degree m
in II. When the difference relation is used for successive values
of fj,, we obtain expressions for the successive coefheients c, which
give each of them as a multiple of c^ by a quantity that is a
rational function of fi. When these coefficients are used, we have
the formal expression of a quantity vj which satisfies the equation
Unfortunately for the establish men t of the regular integrals, this
formal expression does not necessarily (nor even generally) con
verge: for, in the differencerelation among the constants c, the
righthand side is a polynomial of degree m in [i, while the left
hand side is a polynomial of degree m — m in /i, so that the series
Se^a^'''''
would, as in the preceding investigation, generally diverge.
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79.] EEOULAR INTEGRALS 239
But while this is the fact in general, it may happen that the
series would converge when p acquires a value occurring as a root
of the equation
£(p) = 0.
In that case, the series satisfies the equation
if(»)0;
in other words, it is a regular integral of the differential equation.
Further, if the particular value of p be a multiple root of the
indicial equation, it can happen that the series
div
Tp
converges for this particular value of p ; and then
^g).lo.«(p).)
= 0,
because the value of p is a multiple root of E= ; in other words,
p is then a regular integral of the differential equation. And so
possibly for higher derivatives with regard to p, according to the
multiplicity of the root of E=0.
The whole test in this method is therefore as to whether the
series
converges for the particular value (or values) of p given as the
roots of the indicial equation. The method of dealing with a
repeated root of the indicial equation has been briefly indicated.
Corresponding considerations arise, when £" = has a group of
roots differing among one another by integers. In fact, all the
processes adopted (in Ch. iii) when all the integrals are regular,
are applicable token only some of them are regular, provided the
various series, whether original or derived, are converging series.
The deficiency, that arises through the occurrence of divei^ing
series, represents the deficiency in the number of regular integrals
below m — n. As already stated, the tests necessary and sufficient
to discriminate between the convergence and divergence of the
various series are not given in any explicit form, that admits of
immediate application.
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240 EXAMPLES [79.
£x. 1. Consider the equation
constructed in § 78, Ex, 1. The indieial equation is
p3=0,
BO that there ia not more than one regular integral ; if it esists, it belongs
to an exponent 3. To determine the existence, we substitute
in the original equation ; that it may he satisfied, we must have
0=^{(» + 2)(™ + l)2}c„_, + nc„,
for « = 1, 2, .... We at once find
and therefore
The series S c„a^*" diverge^, and tlierefoie the one possible le^ulii lute^iil
does not exist; that is, the oiigmal equation po^isc'ises no leguUi intPoHl,
although the indieial equation is of the first degiee
If there were a regular integral, it would satisfy an equation
where w is a holomorpbic function of ^ ; and the original equation could then
be written
(''£) ('i').
where » is some holonsorphic function in the vicinity of x=0. It might be
imagined that, as the indieial equation is of d^ree unity (a property that
does not forbid the existence of a regular integral), it would be possible to
obtain the regular integral through a determination of u, and that the
divergence of the series in the preceding analysis is due to the operator
which annihilates only expressions that are not regular. That this is not the
case may easily bo seen. We have
so that, if the resolution be possible, we have
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79.] EXAMPLES 241
Substituting in the second of these the value of v given by the first, we find
as an equation to determine «, supposed a holomorphio fuiiotion of x. Let
>>e substituted ; in order that the equation for v, may be satisfied, we have
and, for values of m higher than zero,
(K + 2aol)ii, + a(a^a,_i+asa,._s+...)a„ + ,=0.
Hence Oo=3, a,=4, a^=2i, and so oo. The relation giving «„+„ when taken
for successive values of n, shews that all the coefficients a are positive ; hence
>(« + 5)<i„,
that is,
30a„>(™ + 4)!,
and so the series for m diverges : in otber words, there is no function m, and
the hypothetical resolution of the equation is not possible.
Mole. This ai^umont is general ; it does not depend upon the particular
coefficients for the special equation that has been discussed.
Sx. 2. Consider the equation
which is in the normal form. The characteristic index is 1 ; the indicial
equation is
(9(fll)5d+9=0,
{fl 3)2=0,
BO that the number of regular integrals cannot be greater than two, and such
as exist belong to the exponent 3.
To detei'mine these regular integrals (if any), we adopt the Frobenius
method of Ch. Iii, Taking
^=c„xP + c,afi*'^ + ... + c^sf + " + ...,
provided
_ p22p5
■^1 '''* p2 '
and, for values of n greater than unity,
= (p + «.3)c,+{p^ + p(2n4) + )i24H2}c„.i2(p + n)c„_a,
a factor p + m — 3 having been removed, because it does not vanish for these
values of ?i. Let
(p+»3)e,2«„_, = 4„.
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242 DETERMINATION OF REGULAR INTEGRAL [79.
X;. = (p2)<!,3o„=(p3)(p + l)c„.
Also the differenceequation for the coetKeients c becomes
i:,=l^l)'>{p + n)ip + n~l)...{p + 2)k,
=.(_!)» (p_3){p + l)(^ + 2)...{p + ™)c„
Hence, writing
2"
"""nO. + ftS)'*''
in the rtslation
and substituting the value of i,„ we have
Adding the aides of tllis equation, talsen snccessiveiy for m, b1 3, 2, and
noting tliat
.J{5 + »pi>")n()>3)t„
...,.[M5+2,,')n(p3) + (,3)J_(i)."Jsi»yiisi"^].
We thus have a value of y in the form
where
S (irn(p + m)n(p+m4)
:^2".(5 + .pp^) 'V;^ +(p3)^"
n(p3) _ _
nMn(p+»a)
and this satisfies the relation
DV=c„{p~3fxi:
It is clear that formal solutions of the original differential equation ai
'^ [f],.
Of these, the first is
in effect, a constant multiple of sfl^ ; and the second is
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79.] WHEN IT EXISTS 243
because a series, in which
is the coefficient of j^*^, manifestly diverges*.
It thus appears that, although the indicial equation for x=0 is of the
second degree, the differeutial equation possesaes only one integral which is
regular in that vicinity ; and this integral is a constant multiple of x^e^.
This regular integral satisfies the equation
«*(3+»),.0,
SO that the original equation must be reducible. It is easy to verify that it
can be expressed in the form
, + .(3 + 2.)}{.*(3 + 8.);,}.0.
JH.v, 3. As an example which allows the convergence of the series for the
regular integral to occiur in a diffei'ent way, consider the equation
^y'_(l_2a;42a^)y+(I2jr;+;i;2)y=0.
The indicia! equation is
P=0,
so that one regular integral may exist. To determine whether this is so or
not, we substitute
which (if it exists) belongs to the exponent aero. Comparing coefficients,
and, for all values of n that aro greatei' than unity,
(B + I)«„^.i = (»^ + m + l)o„3aa„_i + a„_j.
Let
In general, the values of a (and the consequent values of a) as determined by
the last equation, lead to diverging series ; but in our particular case,
o that 0^=0, 0^ = 0, and generally Cm^O, that is,
• The series in ijg is saved from divergence because, in it, these ooeffioieiits are
muldplioii by the taotor p  3, which vanishes for the special value of p and whidi
therefore removes the quantities that cause the divergence in the Eecond. integral.
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244
and therefofe
EXISTENCE OF
[79.
provided
so that a regular integral esiats. It is a constant multiple of e'.
Ex. 4. Consider the equation
i) (J.) = (aH5 4 ^k") y"" + (^ + 4iK«+ i!*) y'"  (2ii72 + 3j!»+ ar«) y
+ (3ar + toH 4iJ^) y  (3 + to + 4a;S) ,?/ = 0,
The characteristic index is unity ; hence the numher of regular integrals is
not greater than three. To determine them, if they exist, we take an
cxpi'eBsion
and form 2)(y), choosing relations among the coefficients c such that all
terras after the first in the quantity D (y) vanish. We thus find
Cip2(p2) + c„(pl)(p2)(f,Hp3) = 0,
and, for values of n greater than unity,
t.(p + »ir'(j. + »3) + <.,(p + — 2)(p+«3)& + »)'(p+«)3)
+ «.,(, + l>S)'(p + »4)(p + »)_0.
The indkial equation is
(pl)'(p3).0
of degree 3 aa was to be expected ( = 4 — 1), be use tl
is 1, The roots form a single group; ifaregulir t o
the root 3, it wiii be free from logarithms ; f tw eg 1
belonging to the root 1, one of them may or may t b fr t
and the other will certainly involve logarithms.
Consider the root p=3. As p+K3 thei nish f
values of n, wo may remove it from the differen qu fc
<^„(p+«l)'' + o.^,(p + »2){(p + ™)=(p + «)3}
c,(p + ml)Hc,_,{p + m2)(p + »3) = *„,
we at once find
i„ + (p + )i)i„_i = 0.
We require the value of Jc^. We have, for p = 3.
til gt
i m 1 g thm
J th t th 1 tt
Taldng
*a = 1602 + 6e, = 80Co.
i„ = (l)''(« + 3)(»i + 2)...6ij
= §(~l)"(«.i3)!<r,;
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79.] REGULAR INTEGRALS
ao that, writing
Aa « and «a are positive, it follows that all the
and clearly
80 that the series
divei^s ; and there iis no regular integral belonging to the root 3. Moreover,
the coefficient of o„, being (p+n—l)\ does not vanish when p=3 for any
value of TO ; hence, if two regular integrals exist belonging to the root unity
of the indicial equation, one of them will certainly be free from logarithms.
Consider now the repeated toot p = l. As p + )i — 3 vaniahes for this
value of jj when «.=2, the differenceequation is then evanescent for ji = 2 and
it does not determine r^. For other values of n, the quantity p4«3 does
not then vanish, so that it may be removed. We then have, for values of
m ^ 3, the same form of equation as before, viz,
■^„(p + mI)^ + c„i(p4ft3){(p + »t)^(p + ™)3}
+ c„_,(pf»3)(p + «4)(p+«) = 0.
Also
c,= (pI)(pS + p3)^,
the value p = l not yet being inserted because we have to differentiate with
regard to p. Tlie differenceequation for jj=3 gives
9(311802 = 0,
For values of n.^4, let p = iT2, so that the value of o is 3 ; take m2=m,
ao that the values of m are ^ 2 ; and write
then the differenceequation becomes
6,„(^+ml)'f&™.(<r + m2){(<rtw)^(ff + i«)3!
H6«_2(otm3)(oHm4)(olm) = 0.
Here <r=3, m^2; e2=&o, C5 = 6[:
exactly the same as in the former case
,^[b^a^x+a %  )
with tl o cailier notation it certainly diverges inles^ 6j=0. If 6=0, every
ciefecient vanishes, and the senes itself vanishes As we require regular
mte^rils we shall therefore assume 6^ = that is, i.;=0; and then all the
remaining c effiuenta \inisli so that we h \e
1= .[■^ ■'"^'(pl)(p'+p3)^],
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246 EXAMPLES [79.
au osprossion which is such that
are integrals of the equation
The former ia o^s : one regidar integral thus is
The latter ia
another regular integral in
The original differential equation accordingly has two regular integrals.
S!x. 6. Shew that the equation
haa one integral regular in the vicinity of :i;=0, and express the equation in
a reducible form.
Sx. 6. Shew that the equation
has two regular integrals in the vicinity of x=0, in the form
e", x^ ;
and obtain the integral that is not regtilar.
Ex. 7. Shew that the equation
has no integral, that ia regular in the vicinity of m=<i ; ej[preas the equation
in a reducible form, and thence obtain the integral by quadratures. (Cayley.)
Ex. 8. An equation P — can be e.^pressed in the form
$0 = 0,
where ^=0 has no regular integrals ; can P=0 have any regular integrals ?
Illustrate by a special case.
Ex. 9. Ill the equation
the coefficients P are poljDomials in a: of degree p, and p<.n: shew that it
possesses n—f integrals, which are integral fuuctions of a;. (Poincar^.)
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lUKElJUCIELE EQUATIONS
Existence oe laREUUCiBLE Equations.
80, We have seen that an equation is reducible when it ia
satisiied by one or more of the integrals of an equation of lower
order, in particular, by the integral of an equation of the first
order. The main use so far made of this property has been in
association with the regular integrals of the equation: but it
applies equally if the equation possesses nonregular integrals
that satisfy an equation of lower order. It is superfluous to
indicate examples.
It must not be assumed, however, that every equation is
reducible by another, if only that other be chosen sufiiciently
general. On the contrary, it is possible to construct an irre
ducible equation of any order m, as follows*.
We construct an appropriate characteristic function which, as
is known (§ 76), uniquely determines the equation. Take a poly
nomial in p of degree m, say
let the coefficients of the powers of p be holomorphic functions of
«, not all vanishing when « = 0; and let the function, subject to
these limitations, be so chosen that, when arranged in powers of
X in the form
h {^., p) = h, (p) + xh, (p) + a;'h,(p} +...,
ho (p) is independent of p and not zero, and h^ (p) is of degree m in
p. Then if iV=0 is the equation determined by h(x, p) as its
characteristic function, JV= is irreducible.
Were N reducible, an equation S = of lower order s would
exist such that each of its integrals satisfies N = <i; and then an
operator Q, of order m — s, could be found such that
N=QD.
Wc take Q and D in their normal form ; and so N is in its normal
form. Now
Q{af) = af {^,(p) ( a^{p) +,af^,ip) + ...[,
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248 REDUCIBILITY OF EQUATIONS [80.
the righthand sides of which are polynomials in p of degrees m — s
and s respectively. Then, as in § 76, we have
K ip) = ?„ ip) vi (p) + ?. (p) v« ip + 1)
Now /(„ (p) is a constant, being independent of p ; hence, owing to
the polynomial character of Q {w") and D (a^) in terms of p, the
two quantities ^((p) and ijoCp) £"^e constants. Accordingly, ijo{p + 1)
is a constant ; and therefore the degree of
f.</>)i.(p)+f.(p)i.((>+i)
in p is the degree of 171 (p) or fi (p), whichever is the greater. But
the degree of i)i(p) is not greater than s, and that of ^i{p) is not
greater than m — s; so that, as s > 0, the degree is certainly less
than m. But the espresaion is equal to A, (p), which is of degree
tn. Hence the hypothesis adopted is untenable ; and the equation
N=0, as constructed, is irreducible.
Equations having Regular Integrals are Rgducible.
81, Suppose now that, by the preceding processes or by some
equivalent process, the regular integrals of the equation N=0
have been obtained, s in number, and that the equation of which
they constitute a fundamental system is S = 0, of order s: a
question arises as to the other m — s integrals of a fundamental
system of N= 0. Let
where T and S (and therefore also iV) are taken in their normal
forms. The s regular integrals of if, say y,, y^, ..., ijs, all satisfy
iS = ; and no one of the m—s nonregular integrals of N, say
W], Wj, ..,, Wm—s. satisfies jS = 0, for this equation has all its integrals
regular. Let
S(»,) = »„ (.=1 m,);
then, as N{wr) = 0, we have
r(«,) = o.
Now Wr is not a regular expression; hence n^ is not regular,
that is, it contains an unlimited number of positive and negative
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81.] HAVING HEGULAK INTEGKALS 249
exponents when it is expressed as a powerseries. Accordingly,
the m — s quantities u are integrals of the equation
which is of order m — s and has no regular integrals ; and the m~s
nonregTilar integrals of iV= are given by
S (»,) = «,.
it being sufficient for this purpose to take the particular integral
and not the complete primitive of the latter equation.
The case which is next in simplicity to those already discussed
arises when s = ni — 1, so that the original equation then possesses
only one integral which is not regular. The equation T^O is
then of the first order.
With the limitations laid down, the normal form of T is
d
where q, and q, do not become infinite when a; = 0. As the integral
of T(u) = is not regular, it follows that ^i does not vanish and
that g,, does vanish when a:= 0; so that, if
where a is a positive integer > 1 and Q {x) is a holomorphic
function in the vicinity of « = 0, sach that Q (0) is not zero, the
equation determining u is
1 ^ a. A_ _9i 
say
lg + ^i + ?i<^ + ... + ? + iiWo,
where iJ (a:) is a holomorphic function of x in the vicinity of iC = 0.
This gives
%+ »i, + .. + '
uare' ' P.W,
where Pi is a holomorphic function of x in the vicinity of a: = ;
and then to determine w, the nonregular integral of N—0, we
need only take the particular integral of
yGoosle
250 REDUCJBTLITY [81.
where
in which qa, qi, ..., Jmi denote lioloraorphic functions of x, and ^o
does not vanish. Writing
the equation for v takes the form
where qo, p,, p^, ..., pwti are holomorphic functions of «, such that
5o and pmi do not vanish when iC = 0.
In some cases it happens that a particular integral of this
equation exists, in the form of a converging powerseries repre
sented by
«'— "— PW.
where P{x) is a holomorphic function of a;: in each such case, the
nonregular integral of the original equation is
«■"'— «°PW.
But, in general, the particular integral of the uequation is not of
the same type as the regular integrals of the original equation :
and then the nonregular integral of the preceding equation
cannot be declared to be of that type.
Ex. An illuatration is furniahed by the equation in Bs, 6, § 79, viK.
It has two regular integraJa, viz.
and these constitute the fundamental system of
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81.] THE ADJOINT EQUATION 23
in the normal fonn. To have the given equation in the normal form, v
multiply throughout hy x'' \ and then it must be the same as
when f is properly determined. We easily find that
and so the equation for determining m, where
y being the nonregular integral, is
v.<ix~ 3?{\■\■'i^•\■%a?^^a^)
ao that
Hence the nonregular integral of the original equation <ai'i
particular integral of
„ ^ , 1(2j; + 3j;3 + :k* \
fy^y^— ^ 1".
Let j/ = v^ ; the equation for v ia easily found to be
"(i)
satisfied by v = l : and therefore the nonregular
The Adjoint Equation, and its Properties.
82. Of the properties characteristic of a linear equation, not
a few are expressed by reference to the properties of an associated
equation, frequently called Lagrange's adjoint equation. It is a
consequence of the formal theory of our subject, as distinct from
the functional theory to which the present exposition is mainly
limited, that Lagrange's is only one of a number of covariantive
equations associated with the original. As its properties have been
studied, while those of the others remain largely undeveloped.
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252 PROPERTIES OF lagrange's [82.
there may be an advantage in giving some indication of a few
of its relations to the original linear equation.
The latter is taken in the customary form
P(?^) = P,wi»' +P,wi"'» + P,w"'=' + ... +P„«; = 0,
where w'*"' is the rth. derivative of w with respect to z; and from
among the various definitions of the adjoint equation, we choose
that which defines it to be the relation satisfied by a quantity v in
order that vP(w) may he a perfect differential. Now, on inte
grating by parts, we find
+ ^, (vPr) W l"^^> ...+( 1 )"^ [w ^ {vPr) dz,
for all the values of r\ hence, writing
P. =/•.,
R (w, v) = p„w '""" 4 piW "'~^' + . . . + paiW,
PW = P,.^(P,_„) + ;(P,...)^.., + (1)"£(.P.),
we have
jvP (w) dz = R (w, v) + iwp {v) dz,
and therefore
.P(«.)«p(.).^(K(»,.)}.
It is clear that, in order to make vP{w) a perfect differential,
whatever be the value of w, it is necessary and sufficient that v
should satisfy
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82.]
ADJOINT EQUATION
a linear equation of order n, commonly called Lagrange's adjoint
equation ; and further that, if v is regarded as known, then a first
integral of the equation P (w) ;= is given hy
B {«,.«) = ..
a being an arbitrary constant, and R being a function manifestly
linear in w and its derivatives.
Further, since
jwp (v) d^^R (w, V) + jvP (w) dz,
it is clear that wp {v) is a perfect differential if
P(») = o,
shewing that the original equation is the adjoint of the Lagrangian
derived equation: or the two equations are reciprocally adjoint to
one another.
Ex. Shew that, if ic,, ,.., w„ be a fundamental system of integrals of the
et[uation i*(w) = 0, then a fundamental system of int^rals of the, adjoint
equation p{v) = lS is given by
1 Ir."
Shew also that the product of the respective determinants of the two sets of
fundamental integrals depends only upon 1'^.
One immediate corollary can be inferred from the general
result, in the case when the equation P{w) = Q is reducible.
Suppose that
p(«,)=p.p,(»)=p,(fl'),
say, where W^P^iw); then we have
(vP(w)ds= Lp,(W)dz
^R,(W,v)+jwP,(_v)dz,
■where P, is the adjoint of P„ and R^ is of order in W and in v
one unit less than P,. Again, writing
F=P,W,
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254 THE ADJOINT EQUATION [82,
we have
where P^ is the adjoint of P^, and iE^ ia of order in V and in v one
unit less than P,. Combining these results, we have
jvP(w)dz^R,(W,v)\R,{V,v) + jwP,(V)dz
= R{w,v) + jwP,P,(v)dz,
where R is of order one unit less than P in w and in v. It follows
that
P,P,{v)^0
is the adjoint of
P(») p.p. (") = !),
where P, , Pi are adjoiut to one another, and likewise Pj, Pj.
By repeated application of this result, we see that the adjoint
of
P<«/j=P,P,...P,(M.) =
is given by
P;P^,...P,P,(i')=0,
Hence the adjoint of a composite equation is compounded of the
adjoints of the factors taken in the reverse order. Manifestly
an equation and its adjoint are reducible together, or irreducible
together.
The expression R (w, v) is linear in the derivatives of w, up to
order n — 1 inclusive, and also in those of v, up to the same order :
it may be called the bilinear concomitant* of the two mutually
adjoint equations.
For further formal developments in respect to adjoint equa
tions and the significance of the bilinear concomitant, reference
may be made to Frobeniusf, Halphonj, Diui§, Cels], and
Darbouxt.
* Begleitender hilinearer Differentialaitsdmclc. with Frobeniua,
t CreUe, t. lxixv (167S), pp. 1S5— 213; referencea ace given to other writers.
t Liouville'a Journal, i' S^c, t. i (1885), pp. 11—85,
§ Aim. di Mat., 3» Ser., t. n (1899), pp. 297—324, ib., t. ni (1899), pp. I2S— 183.
II Ann. de VEc. Norm., 3' Sir., t. viii (1891), pp, 341^15.
IT Theorie generale dee surfaces, t. ii, pp. 99 — 121.
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82.] EXAMPLES 255
Ex. 1. Prove that, if ii linear equation of the second order ia selfadjoint,
it is cspressible in the form
that if a linear equation of the third order, in the form
ia effectively the same as its adjoint equation, then
and find the conditions that a
selfadjoint.
r equation of the fourth order should be
Ex. 3, Prove that, if the equations
yoHWinyiVi""'!! "Yi — y.j»ff"~^4=0,
e adjoi
it to one another, then
ri= ffi+ffo'y ^1= 7i + To''
73= ffs + ^ffi^ffi"+go"\ ?3= ■ V3 + 3y2■■V + 7o"'>
and obtain the expression of the bilinear concomitant.
i„ denote any « arbitrary functions of x, such that
dx^''
does not vanish identically; and suppose that these functions of x are
regular in a given region of the variable, aa well as the coefficients a of the
etjuation
Further, let a set of quantities p bo constructed according to the law
dpn dp, dv„_,
»"•• ?■"<£. »'".j;f' ■■•■ '••'•^nr
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256 PROPERTIES OF laorange's [82,
and let the last of them be denoted by  Z, so that there are n functions Z
con'osponding to the n functiona z. Shew that, if Q (c) is the value of § when
the last column of the latter iiS replaced by constants c„ ..., c„, if §{a', a\) is
its value when the last column is similarly replaced by Sj (a:,), %(a:i), ...,
j„ (x^, and if Q {x, a^i) is its value when the last column is similarly replaced
by 2i (:Kj), iTj (^cj), ...,Z„(a;,), then
where a is a value of 3! within the given ri^ion and the constants c are
determined in association with a.
Indicate the form of this result when z^, ..., 2„ are a fundamental system
of the equation, which ia the adjoint of the lefthand side of the above
equation.
Also shew how, in even the most genera! case, it can be used as a formula
! to obtain an infinite converging series of integrals as an
1 for y. (Dini.)
83. Consider an expression Piw) and its Lagrangian adjoint
p (v), and let R (w, v) denote their bilinear concomitant ; then
,P(»)»;,(rt.^j(2i(»,„)),
which holds for all values of v and w. Accordingly, let
w = 2~''~''~^, V — z",
where s ia any integer ; then
^p (,,—.) _ ,,.,^ (^) _ ^ ^M(z>—'. ^)).
Now the lefthand side is a series of powers of z, having integers
for indices ; as it is equal to the righthand side, which is the
first derivative of a similar series of powers, the lefthand side
must be devoid of a term in s"'.
Let
Piz% =SMt)z^^,
be the characteristic function of P {w) ; then the coefficient of z~^
in scP {2^'') is /s ( p  5  1). Further, let
be the characteristic function oip{v); then the coefficient of 3~'
in s"*"^'^ (s**) is 4>»{p) Hence
>^,{(>)=Mpsn
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83.] ADJOINT EQUATION 257
and therefore
so that, if
be the characteristic function of a given equation, then
S/, ((>/" 1)2"'
is the characteristic function of the adjoint equation.
When P{w) is in its normal form, all the coefficients /^(p)
vanish for negative values of fi, but /i,(p) is not zero. Hence
f^{— p — fi — l) vanishes for negative values of p,, but not
/ti(~ p ~^)'} and therefore the adjoint expression p(v) is in its
normal form. Moreover, their indicial functions y, (p), <p^ (p) are
such that
/o(/>)=0o(pi), Mp)=A(pn
80 that they are of the same degree*, or the characteristic
indices are the same. Hence if an equation has all its integrals
regular in the vicinity of a singularity, the adjoint equation also
has all its integrals regular in the vicinity of that singularity ; for
the characteristic index is then zero for the original equation, and
it therefore is zero for the adjoint equation. Similarly, if am
equation has all its integrals nonregular in the vicinity of a
singularity, tlie adjcnnt equation also has all its integrals non
regular in the vicinity of that singularity ; for the characteristic
iudes: is then equal to the order of the original equation, and it
therefore is equal to the (same) order of the adjoint equation.
On the basis of these two results, we can obtain a descriptive
condition necessary and sufficient to secure that, if a differential
equation of order m has an indicial function of degree m~n, the
number of its regular integrals is actually equal to m — n.
Let P=0 be the differential equation, with an indicial func
tion of degree m — n. Let R=0 be the differential equation
of order m — n, which has the aggregate of regular integrals of
P = for its fundamental system ; its indicial function is of degree
ran. Then (§ 75, iv) the equation P = can be <
the form
* Thom^, Crelle, t. lssv (1873), p. 276; Frobeiiiua, Crelk, t
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258 Lagrange's [83.
where Q is a differential operator of order n. Because the degrees
of the indicial functions of P and li are equal to one another, it
follows (from § 76) that the degree of the indicial function of Q is
zero, that is, the indicial function of Q is a constant, and therefore
(§ 77, Cor. i) the equation Q = has no regiilar integral.
Now construct the equations which are adjoint to P = 0, Q = 0,
R — respectively ; and denote them hy p = 0, ? = 0, r = 0,
Because R and r are adjoint, and hecause all the integrals of
R = are regular, it follows that all the integrals of r = are
regular; and conversely. Similarly, because Q and q are adjoint,
and because Q = has no regular integral, it follows that q =
has no regular integral ; and conversely. Further, by § 82, we
have
p^rq,
so that the equation adjoint to P = is
p = rq = 0,
and this equation possesses all the integrals of 5 = 0, an equation
whose indicial function is a constant. Hence it is necessary that
the equation adjoint to P = should possess all the integrals of an
equation of order n, having a constant for its indicial function, if
P = is to have m — n linearly independent regular integrals.
But this descriptive condition is also sufficient to secure this
result. For, as the condition is satisiied, we have
p = rq,
where the indicial function of q is 2. constant ; hence, with the
preceding notation, we also have
P^QR,
and the indicial function of Q is a constant. Accordingly, as the
indicia! function of P is of degree m — m, it follows (§ 76) that the
indicial function of R is of degree m — n; and therefore (Ch. Ill),
as the order of R — is m — n, all its integrals are regular. But
P = possesses all the integrals of P = ; and therefore it has
m~n regular integrals.
We therefore infer the theorem : —
In order that an equation of order m, having an indicial function
of degree m — n, may possess m — n regular integrals, it is necessary
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83.] ADJOINT EQUATION 259
and suffia'piit that the adjoint equation should possess all the inte
grals of an equation of order n, having an indicia! .function which
is a consUvnt
This result was first established by Fiobenius*; and it may
be compared with the corresponding result obtained by Floquet
(§ 77). The special case, when w = 1, had been previously discussed
by Thom^f, who obtained the result that an eqtuition of order m,
having am, vndiciat function of degree m — 1, possesses m — 1 regular
integrals, if the adjoint equation has an integral of the form
/©_!».«,
where G(] is a polynomial in  , and a is a constant.
We shall not pursue this part of the formal theory of linear
differential equations further : we refer students to the authorities
already (§ 82) quoted, as well as to Thom^j, Floquetg, and
Griinfeldil.
* Grdle, t. lxks (1875). pp. 331, 332.
t Crelle, t. lisv (1873), pp. 278, S79.
X A summary of many of the memoirs a
Thomfi, published in CreJie's Journal, will t
pp. 185—281.
% Ann. de VEc. Norm. Sup., 2' Ser.. t. viii (1879), Supplement, p. 132.
II CTelle, t. oxY (1895), pp. 328—842, ib.. t. oxvii (1897), pp. 278290,
ib., i. cxiii (19tX)), pp. 4352, 88.
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CHAPTER VII.
Normal Integrals; Subnormal Integkai^.
84. It is now necessary to consider those integrals of the
differential equation in the vicinity of a singularity, which are not
of the regular type. Suppose that such an integral, or a set of
such integrals, is associated with a root 9 of the fundamental
equation (§ 13) of the singularity which, as in the last chapter,
will be transformed to the origin by the substitution
1
z — a = x, z — ,
according as it is in the finite part of the plane, or at infinity.
Let p denote any one of the values of
then it is known that an integral exists in the form
where i^ is a uniform function of x in the vicinity of the origin.
As this integral is not of the regular type, the function ^ will
contain an unlimited number of negative powers, so that the origin
is an essential singularity of ^ : in the case of the integrals con
sidered earlier, the origin was either a pole or an ordinary point.
Accordingly, when ^ is expressed as a powerseries, it will contain
an unlimited number of negative powers: it may contain an
unlimited number of positive powers also, and in that case it has
the form of a Laurent series.
Classification of such integrals might be effected in accordance
with a classification of essential singularities ; but the discrimina
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84.] NORMAL INTEGRALS 261
bion that thus far bas been effected among essential singularities
is of a descriptive type *, and has not led to functions whose general
expressions are characteristic of various classes of singularities.
Accordingly, it ^ possible to choose one function after another
with differing forms of essential singularity, and to construct
(where practicable) the corresponding linear equations possessing
integrals with the respective types of singularity : but there is no
guarantee that such a process will lead to a complete enumeration.
There is one such function, however, which is simpler than any
other, and yet is general of its class. It suffices for the complete
integration of the linear equation of the first order when the origin
is a pole of the coetScient ; and an indication has been given (§ 81)
that it may serve for the expression of an integral of an equation
higher than the first. The equation of the first order may be
taken to be
where a^+'P is a holomorphic function, s being some positive
integer. Let
where /' (ic) is a holomorphic function ; then we easily have
where "^ {x) is a holomorphic function of w, and
n = 2+ii+ •■• + ■
a!* a^ ' X
It is clear that ic=0 is an essential singularity of the integral;
and also that we thus have the complete primitive of the equation
of the first order.
It appeared, in § 81 and the example there discussed, that
such an expression, if not in general, still in particular cases, can
be an integral of an equation of higher order.
As all expressions of the form
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2(i2 THOMfi'S NORMAL [84.
where il is a polynomial in  , possess the same generic type of
essential singularity, we proceed to the consideration of equations
that may possess integrals of this form. Such an integral is
called* a normal elementary integral or (where no confusion will
occur) simply normal. The quantity e", through the occurrence
of which the point a) = is an essential singularity, is called the
determining factor of the integral; the other part of the integral,
being ie<'iy {x) where i^ is holomorphic, is of the type of a regular
integral, and so the quantity p is called the eivponent of the
integral.
Construction of Normal Integrals.
85. We proceed, in the fii'st place, to indicate Thome's
method f of obtaining snch normal integrals as the equation
d™ti! d"^^!/) dw _
die™ '^ dx™~^ ' ' ' "™~^ ^(c "™
may possess. (The method gives no criteria as to the actual
existence of normal integrals: and therefore, if any criteria are
to be obtained for equations of order higher than the lirst, they
must be investigated otherwise.) If a normal integral exists, it
is of the form
where ii is a polynomial in  ; and il is determined so that, if
possible, the equation satisfied by u may possess at least one
regular integral. Lot
ftc"
so that
(,= 1, *, = n', tp^,^tp' + Q.'tp, {p=l,2, ...);
then
d"w _ /, , du , d"~'u ^ d"M\
dx" \ die dx" ' dafJ
* Thom^, Crelle, t. sov (1883), p. 75. Cajley, ib., t. c (18S7), p. 286, suggeated
the came suinegular ; but the name normal is that nhioh has geiierall; been
adopted.
t Crelte, t. lsxvi (1873), p. 292.
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85.] INTEGRALS 263
When these quantities, for the successive values of n, are substi
tuted in the differential equation for w, the determining factor e"
can be removed ; and the differential equation for w then is
dt. d—'u
du
, , (".D!
„, , ("'2)> ,
■)! ' (rl)!(mi
■)!"'"' (r2)!(>ni)!'
...+(»tr + l)p.
forr = l, 2, ...,m.
If the origioal equation possesses a normal integral, then, after
the proper determination of fl, the differential equation for u will
possess at least one regular integral : its characteristic index
cannot then be greater than m — l, which (after the results in the
preceding chapter) is a necessary but not a sufficient condition.
As 12 is a polynomial in «', its form and degree being un
known, let its degree be s — 1, so that s^2; we then have for £1'
an expression of the form
il'
. «£ , «3 ,
Hence in ti, the governing term (that is, the term with highest
negative exponent of a:) is ^ ; in ij, it is J^ ; and so on, so that, in
*„, it is ^. As in § 73, let ot, denote the multiplicity of « = as
a pole of p, ; then in q^, the governing exponents of its respective
parts are
rs, ^, + (rl)s, ■=r, + (r2)s, ..., ^ri + s, ^,.
Thus the governing exponents in q,. are, so far as they go, less than
those in q^+i by s, and s ^ 2. Hence, in forming the characteristic
index for the equation in u, for the purpose of determining whether
it may possess a regular integral, the governing exponent in q^
is certainly greater by s than that in any otiier coefficient; the
characteristic index is m, the indicial function is a constant, and
the equation has no regular integral. But, thus far, ii is quite
arbitrary ; and it may be possible, by proper choice of its constant
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264 DETERMINATION OF [85.
coefficients, to secure that a number of the terms in q^ with the
greatest exponents of *■"' shall disappear. If by thus utilising the
governing exponent and the constants in SI', we can secure that
the characteristic inde\ of the equation in u is less than m, the
indicial function ceases to be i con^tint and the equation may
have a regular intejfral
In Older that the indicial function may not be a constant, the
governing exponent of q , must be less than that of q^, by unity
at the utmost, or that of q,^ must be less than that of q^ by two
at the utmost, or (tor sjme value of }) the governing exponent of
qmr must be less than that of 5 by ? it the utmost ; whereas at
the present moment, these diminutions are s, 2s, rs respectively,
where s ^ 2. Hence an initial necessity is that the s — 1 terms in
qm with the highest exponents of a:'^ shall vanish. Now
qm,^tm + Pltml + ■■■ +Pm~A+l^m
The s — 1 terms in t^ with the highest exponents ol' le"' are the
same as in II'", because of the form of il' and because
(but not more than those s — 1 terras are the same) ; hence the
s~l terms with the highest exponents of (e~^, say the first s — 1
terms, in
n'"' +piii''"~' + . . . +p„jfi' +p„
must vanish.
86. To render this result attainable, it is necessary that the
greatest exponent must not occur in only a single term of the
preceding expression, for then the term could vanish only by
having d, = ; the greatest exponent must occur in at least two
terms. Consequently no one of the numbers
ms, ■=T, + (m — l)s, ■5r2 + (m~2)8, ..., ot„,
may be greater than all the rest, that is, no one of the numbers
0, ^i~s, W3 — 2s, .... vT^ — ms,
may be greater than all the rest. Of the quantities
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86.] NORMAL INTEGRALS 265
let g be the greatest. Evidently g is greater than unity ; for the
original differential equation has not all its integrals regular, and
so CT„ > w for at least one value of n. Now s cannot be greater
than g\ for any such value would make all the integers in the
series
0, ^i — s, «;,— 2s, ..., Wmms,
negative except the first, that is, the first would then be greater
than all the rest. Hence s^g: and g ^ 2, from the nature of the
I. When g <2,no value of s is possible; and then there is no
normal integral of the type indicated. Such a case arises for the
equation
when p and q are holomorphic in the domain of ic = and neither
vanishes when a; = 0. The quantity g is the greater of 1, , that
is, it is less than 2 ; so that there is no normal integral. Moreover,
as the indicial function of the particular equation is a constant, it
has no regular integral.
II. When g is an integer (necessarily greater than unity), wc
manifestly might take s = ^. For two at least of the numbers
would then be equal to the greatest among them, which is zero ;
and then two at least of the numbers
ms, JiTj + (m ~ 1) S, Wa + (jrt — 2)s, ..., ■nr^_i + S, CT^,
would be equal to the greatest among them, one of these being ms.
More generally, let n be the characteristic index of the original
equation, so that
for all values of /t that are greater than w; then, adding {m—n)(«—l)
to each side of the inequality, we have
■OT„ + (m  n) s^w^ + imii^s + ifi n)(s  1),
where fi>ii. In the case of all these numbers, {/j. — n){s—l) is
certainly positive ; so that the first s — 1 terms in our expression
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266 NOEMAL INTEGEALS [86.
are not affected by the quantities coiTesponding to a^i^ + im — )j.)s,
and they can occur only through the quantities corresponding to
w^ + (m  X) s,
for X= 0, 1, ...,n, where T>r„ = 0, and n is the characteristic index
of the original equation. We thus consider the first s ~ 1 terms in
and this holds for any value of s equal to or greater than two.
As regards g, which is the greatest among the quantities
it occurs only among the first n, in the present circumstances ; for
it certainly is greater than unity and if any one of the last m — n,
(say — liT^ is the greatest of these last m — n), is greater than unity,
then because
we have
n /J \li /\n 1
that is,
for /i is greater than n. Thus g does not occur in the last m — ii
of the quantities, if one or more than one of them is greater than
unity ; and it certainly does not occur among them, if no one of
them is greater than unity. Hence g is the greatest among the
quantities
It may occur several times in this set ; let  tct, be the first occur
rence, in passing from left to right, and  ^^ be the last. Take
first s = g\ then we have
■^^■\{wi, — k) h = mg, OT,. •\{rn, — r)s = vig,
■wx + (m  \) s < mg, if \ < k, or if X > r ;
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8(5,] THEIR DETERMINING FACTOR 267
SO that the highest terms of ail, being those with index Trig,
occur in
ii'", p,ii'"~ + ... +prn.'«':
K then
p, = x''^{c, + d„a; + ...). (,7 = 1, 2, ...),
the equation which determines a^, the coefficient of a:~i' in il', is
o/ + c^a/' + . . . + c,. = 0.
The remaining g — 2 coefficients in il' are given by equating to
zero the coefficients of the next ^ — 2 terms in
fl'" + piil'"' + . . . + p„_iii' I p„.
Each set of values of the coefficients determines a possible form
of Xi' and therefore a possible form of determining factor. The
number of sets, different from one another, is ^ r.
The preceding cases arise through s = g; but if g, being an
integer, is greater than 2, other values of s, less than g, may be
admissible. They can be selected as follows*. Mark the points
0, n; a,, nl; wj, n  2 ; ...; «r„, 0;
in a plane referred to two rectangular axes; and taking a line
through the first of them parallel to the axis of a;, make it swing
round that point in a clockwise direction, until it meets one or
more of the other points ; then make it swing in the same direc
tion round the last of these, until it meets one or more of the
remaining points ; and so on, until the line passes through the
last of the points. There thus will be obtained a broken Hue,
outside which none of the marked points can lie.
If a line be drawn through any of the points, say sj,, n — v, at
an inclination tan"';t to the negative direction of the axis of ^, its
distance from the origin is
(i+^r'K+(»«>*'l,
so that, for a given direction /a, the distance is proportional to
It therefore follows that an appropriate value of s is given by any
portion of the broken line, which is inclined at an angle tan~"'/i to
' Thy method is due to Fuiseux; see T. K, S 96.
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NORMAL INTEGRALS
the negative direction of the axis of y, where ^ i
integer, ^ 2 : the value of s being
As many values of s are admissible as there are portions of the
broken line with inclinations tan~^/(., where /4 is a positive integer,
which is > 2.
For each admissible value of s, arising from a portion of the
broken hne, the terms in
which correspond to the points on that portion, give the tertns of
highest negative power in x. If, for instance, a portion of line,
having as its extremities the points corresponding to
p,Q'"~' and p(il'"~', {t>r),
gives a value g' (necessaiily an integer, as being a value of s), then
the coefficient a^ satisfies an equation
Cra/~'' + ... + c, = 0,
and the remaining ^ — 2 coefficients in li' are obtained in the
same manner as before. Each set of values of the coefficients
determines a form of li' and therefore also a possible determining
factor; and the number of sets different from one another is
^t — r.
And so on, with each piece of broken line that provides an
admissible value of s.
III. When the greatest of the quantities
is greater than 2 but is not an integer, we construct a tableau of
points as in the preceding case, and draw the corresponding line.
Only such values of s (if any) are admissible as arise from portions
of the line, which are inclined at an angle tan~^ p. to the negative
part of the axis of y, fj. being an integer > 2,
87. In every case, where a possible form of il' and thence a
possible form of fl have been obtained, we take
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87.] SUBNORMAL INTEGRALS 269
If a normal integral of the original equation exists, the equation
for M must possess a regular integral ; and each regular integral of
the latter determines a normal integral of the former having the
determining factor e". An upper limit to the number of integrals
thus obtainable is furnished by the degree of the indicial function
of w ; but the investigations of the last chapter shew that, when
the degree of the indicial function is less than the order of the
differential equation, the number of regular integrals may be less
than the degree and might indeed be zero. The simplest mode of
settling the matter is to take a series of the appropriate form,
determined by the indicial function of the wequation, substitute
it in the differential equation, and decide whether the coefficients
thence determined make the series converge. The normal
integral exists or is illusory, according as the series conveiges
or diverges.
When the normal integral exists, we say that it is of grade
equal to the degree of il as a polynomial in ar^.
Subnormal Integrals.
88. In the preceding investigation of normal integrals, it was
essential that the number s should be an integer ^ 2 : and
accordingly, such values of j/., as were given by the Puiseux
diagram and did not satisfy the condition, were rejected. But
though they are ineligible for the construction of normal integrals,
they may be subsidiary to the construction of other integrals.
Let /t denote such a quantity, given by the Puiseux diagram
in the form of a positive magnitude that is not an integer: its
source in the diagram makes it a rational fraction which, being
expressed in its lowest terms, may be denoted by hik. The
terms which, for this quantity as representing a possible degree
for il', have the highest index of (ir' in
Il'« + piil'"^' + ... +p„,.ri' +p,„
are those which corr^pond to points on the portion of the line
that gives the value of /t. Hence, taking
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270 SUBNORMAL [88.
an equation is obtained by making the aggregate coefficient of
this term of highest order disappear ; the equation determines A.
Now take a new independent variable f such that
and make it the independent variable for the differential equation ;
dD. A
ao that
and therefore
h — k ^
Thus ii is infinite when a: = 0, provided h>k, that is, for values of
jj. that are greater than unity. Accordingly, when we proceed to
consider the differential equation with ^ as the variable, values of
fi of the preceding form can be obtained by the earlier method :
in fact, we may obtain a normal integral of the equation in its
new form, the conditions being that the equation for v. which
results from the substitution
shall have a regular integral or regular integrals. When once the
value of k is known and the transformation from a; to ^ has been
effected, the remainder of the investigation is the same as for the
construction of normal integrals of the untransforraed equation.
Examples will be given later, shewing that such integrals do
exist. As they are of a normal type in a variable x", where k is a.
positive integer, they may be called subnormal'^. Their existence
appears to have been indicated first by Fabryf,
89. We have seen that, if g denote the greatest of the
quantities
^x, W^. \^,, ...,
* Poiiicare, Acta Math., i, viii, p. 304, ealis them anormaUs.
+ Sar Us intigrahs du iquations di^'erentieiiea linSaires b, coefficients rationels,
jThfise, 1883, GauthierViUttra, Paris), Section iv.
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89.] INTEGRALS 271
and if the equation possesses a normal or a subnormal integral of
the form
then il' is a polynomial in ^~' (or in some root of 3~') of order
equal to or less than ff ; and therefore il is a polynomial in s~^ of
order equal to or less than g — 1 Let
then Jt is called the rank of the differential equation for s—0.
When li is an integer, the grade of a normal integral may be
equal to M: if not, it is less than B. When R is not an integer,
let p denote the integer immediately less than R ; the grade of a
normal integral may be equal to p or may be less than p. When
k
R Ts 3, fraction, equal to : when in its lowest terms, then a sub
normal integral may exist having a determining factor e", where
fl is a polynomial of degree k in z ' ; it will still be said to be of
grade j in s, that is, of grade R. All subnormal integrals are of
grade R or of grade less than R.
Ex. Olitaiii t)ie rank of the equation
for 2 ^ CO , the coefB.oiBiits p being polynomials in £.
90. The converse proposition, due^ to Poincare, is true as
follows : —
If n normal or subnormal functions are of grade equal to or
less than B, and have the origin for an essential singularity, they
satisfy a linear differential equation of order n and rank not
greater tha R fa
Any n f net ns "^at sfj ■i 1 r I tt rentiai equation of order
n : in the j sent case let tie
^+P ^ ■+ +PnVJ = 0.
a a
* Ada Ml 8 p 30 ii s been somewhat altered, so as
to admit the □ ma and n a tegrals together.
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POINCARfi'S THEOREM ON
[90.
Let the normal and the subnormal functions be arranged in a
sequence of descending grade: when so airanged, let them be
so that, if S,, i^a, ..., Rnhe their respective grades,
R ^ ill ^ Ra ^ ■ ■ ■ ^ Rn— ! ^ Rji— 1 ^ B«
a the fundamental determinant of the n functions, viz.
Now
d"'
and A„,y is obtained from A by substituting the derivatives of
order n for the derivatives of order Ji — ?• in the rth row. The
value of P,. is
In order to obtain the degree of s = as an infinity of P,., it will
be sufficient to consider only the governing terms in A and A,(^r;
and the degree is determined through the differences between the
two sets of most important terms in the rth rows. Now if
d'iwp
= 0. We taku out of
where 0g_j, is finite (but not zero) when i
the pth column a fector
for each of the n values of y; we take out of the mth row a
factor
for each of the n values of m ; and then every constituent in the
surviving determinants A an<i A„,r is finite. The initial terms in
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90.] NORMAL AND SUBNOKMAL INTEGRALS 273
these constituents are the same for all the rows except the
(r — l)th: the difference there is that ai", Oa", ..., a„" pccur in A'„,ri
while ai""', Oj""^, ..., ««""*■ occur in A', where A'„_, and A' are the
modified determinants, and Oi, Og, ..., a„ are the coefficients of the
governing terms in fi,, fl^. ■^ ^n Accordingly, if
A' =As^+...,
then
A'„,r = ^V4...,
where the other indices are higher than $, and A, A' are constants;
and therefore
i ^ 3ininii (Bi+ii eSOp a^iTp ^.^
the summation in the exponents heing for values I, 2 n of p.
Hence
Now A, being the fundamental determinant, does not vanish
identically : and as ^ = is an essential singularity, and not
merely an apparent singularity, A does not vanish when s = 0;
thus A is not Jiero. It might happen that ^' = 0; but in any
case, if ST, denote the order of 2 = as a pole of the coefficient P,,
we have i^,.Kr(Ri+l}. Thus the largest of the numbers z^r
is <.R,+ 1 ^ R + l ; and therefore, for z = 0, the rank of the
equation < R, which proves the proposition.
When all the integrals are normal, which is the circumstance
contemplated by Poincare, the quantities R are integers and the
determinants A', A'„^, are uniform ; so that the coefficients P then
are uniform functions of z. The coefficients P are uniform also
when the aggregate of subnormal integrals is retained : the proof
of this statement is left as an e
Note. An equation, which has a number of normal integrals,
is reducible; so also is an equation, which has a number of sub
normal integrals.
By the preceding proposition, the aggregate of the normal
integrals (or of the subnormal integrals) satisfies a linear equa
tion with uniform coefficients, say JV = 0, of which they are a
F. IV. 18
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27i EXAMPLES OF [90.
fundamental system. Denoting the original equation by P = 0, we
can prove, exactly as in § 75, that P can be expressed in the form
where Q is an appropriate differential operator. In other words,
P is reducible.
The investigation of the detailed conditions, imposed upon
the form of P by the possibility of such reducibility, will not
be attempted here.
Further, it must not be assumed (and it is not the fact) that
retlucible equations ai'e limited to equations, which have regular,
or normal, or subnormal integrals.
Ex. 1. Consider the equation
whore p, q, T are holomorphic functions of a: that do not vanish when ^ = 0.
To investigate the possible kinds of determining factor, we form the
tableau of points
0, 3 ; 3, 2 ; 5, 1 ; 7, ;
and then eonstruGt the broken line. There are two pieces ; one gives ;< = 3,
the other fi = 3 ; the former joins the first two points ; the last three lie on
the latter. The possible ospreasiona for O' are therefore
where a and 3 ^''S uniquely determinate, and y is the root of a quadratic
equation.
Of course, the actual existence of normal integrals depends upon the
actual forms of p, q, r.
Ex. 2. Shew that the equation
y"'+^?'y"+^ »'+^ '3'=''
where p, g, r are holomorphic functions of ,r that do not vanish when si=Qi,
possesses no normal integrals in the vicinity of j; = G: but that it may
possess subnormal integrals.
Ex. 3. Consider the equation
which has no regular integral, because the indicial function is a constant.
The numbers s'j, org, ctj are 1, 2, 6; so that j — 2, and we therefore take
<=3, so that
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90.] NORMAL INTEGRALS 275
We have to make the single (sl) highest power of a.'~' vanish, in the
eipansion of
in ascending powera of x ; hence
so that a is a cube root of unity, and
Q = ".
Accordingly, we write
after reduction, the equation aatisfied by u is found to be
,„ _ 3M"(a^+6aa^)
The indicial equation for x=0 is
a'(8l).0,
which has a single root 3 = 1 ; so that the wequation possibly may possess a
single regular integral which, if it esiats, will belong to the exponent 1, and
so will bo of the form
As a matter of fact, the wequation is satisfied by
as may easily be verified ; and thus the original equation possesses a normal
integral
where a is a cuberoot of unity. But a may be any one of the three cube
roots of unity ; and therefore the original equation in y possesses the three
normal integrals
.^(^+«^), ^i^+a^^^), e^(x+a:,),
where <i is now an imaginary cuberoot of unity.
The singularities of the equation given by l+6i^ = are only apparent
(§45).
jEe. 4. Prove that the equation
has, in the vicinity of x^a^, two linearly independent normal integrals,
provided a. is of the form pip + 1), where ^ is an integer ^0 ; and obtain
18—2
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276 NORMAL INTBGEAIS [90.
Bx. 5. Prove that each of the equations
aY' + ars/  («s + 2*2) 3^ = 0,
has, in the vicinity of ;i;=0, two hoearly independent normal integrals ; and
obtain them.
Ex. 6. Prove that the equation
has, in the vicinity of ^ = co, three linearly independent normal integrals;
and obtain them.
Ex. 7. Prove that the equation
possesses one normal integral in the vicinity of x = ; and that one normal
integral is illusory in that vicinity.
Ea:. 8. Shew that the equation
(x + %)3fiy"'^{x'iZx'i)^'{a:\%)xy.{1ix'^5x^)y==Q
posaeaaes three normal integrals in the vicinity of x=Q.
[They are iee" , xe " , xe "loga^.]
Ex. 9. Prove that a solution of the equation
is expressed by
where
n' = a^ib, m(X + l) = a(,7 + l)^c.
(Math.,Trip., Part i, 1896.)
Hamburgek's Equations.
91. The conditions, sufficient to secure that an equation, of
order in and not of the Fuchsian type, shall have a regular
integral, have not been set out in completely explicit form
(§1 78, 79) ; and consequently, the conditions sufBcient to secure
that such an equation shall have a normal integral have not been
set out in explicit form. The foregoing examples (§ 90) afford
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91.] hambueger's equations 277
illustrations of the detailed process of settling such questions in
individual instances ; and the following investigation* gives the
appropriate tests for a particular class of equations, which afford
an illustration of the general method of proceeding.
We consider the equation
„ _ a + Ss + j^
1 which a must be different from zero (§ 86) if the equation is to
s a norma! integral. For any integral that occurs, 3 = is
singularity. For large values of ^, the integrals are
regular ; and a fundamental system for s= co is composed of two
regular integrals, which belong to exponents — p, and — pa arising
as roots of the quadratic equation
p{p\} = y.
These two regular integrals may be denoted by
where Pj, P^ are converging powerseries. As the origin is the
only other singularity of the equation (and it is an essential
singularity), it follows that P, and Pj have a = for an essential
singularity ; all other points in the plane are ordinary points for
P, and P^.
The expression of a uniform function having only a single
essential singularity, say the origin^.aud no accidental singularity,
is known by Weierstrass's theoremf to be of the form
where P ( J is a uniform function having all the zeros of the
original function (the simplest form of P being admissible), and
ff ( ] is a holomorphic function of  which is finite everywhere
except at 2 = 0.
* It ia due to Harabui^er, CrelU, t. an (188S), pp. 238—273.
+ T. F., § 52.
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278 SPECIAL EQUATIONS WITH [91.
The function g may be polynomial or it may be transcendental ;
the discrimination depends upon the character of the origin as an
essential singularity for the original function. As the present
application is directed towards the determination of normal inte
grals, the function g (  J will be taken to be a polynomial in  .
If the original function has an unlimited number of assigned
zeros in the plane outside any small circle round the origin, P
is transcendental. When the number of zeros is limited, P[)
is a polynomial in  , which can be taken in the form
where A; is a finite positive integer, / is a polynomial in s of degree
not greater than k, its degree being actually k when ^ = oo is not
a zero.
The equations to be considered are those which have integrals
...p,(l), ^p,(i).
as above, one (or both) of the functions P, and Pj having only a
limited number of zeros outside any small circle round the origin,
with the further condition that the essential singularity at the
origin is of the preceding type. Thus an integral is to be of the
form
say, where il is a polynomial in  , the exponent o is a constant,
and /(z) is a polynomial in z; and the differential equation for u
is to have a regular integral which, except as to a factor z", is to
be a polynomial in z. Let
then the equation for u is
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91.] NORMAL INTEGRALS 279
After the earlier explanations, it is clear that we must take
The equation for u then is
" 2a , , 2a 0 ye
0,
which is to have a regular integral of the type
u = .'/(,)
= z''(c„ + c,z+ ■■■ + C„3" + ...),
there being only a limited number of terms on the righthand
side. The indicial equation for s = is
 200 + 2a  /3 = 0,
BO that
■2a
Substituting the expression for u, and equating coefficients, we
have, after a slight reduction,
{(n\a)(n + ^l)^y]c, = {2a(n + <r} + ^]c^+,
= 2a(n+l)c„+,;
and therefore
(n + a) (n + <rl)'y
'"■'+' 2«(h+1)
It is clear that, if the series with the coefficients c were to be an
infinite series, it would diverge and the integral would be illusory.
For this reason also, as well as by the initial condition, all the
coefficients from and after some definite one, say after Ck, must
vanish ; and therefore we must have
or substituting for er its value, we see that the qi.iadratic equation
whff)e o? = a, m.iist have a positive integer {or zero) for a root.
This condition is sufficient to secure the significance of the series,
and therefore sutScient to secure the existence of a normal integral
of the equation
" _ « + /3 s + "/g'
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280 A CLASS OF EQUATIONS [91.
Clearly, there are two values of a. If for either value the
condition ia satisfied, there is a normal integral of the form
where a has .the value for which the condition is satisfied.
The condition cannot be satisfied for both values, if the values
of <r are different, and arise from different values of p ; for if it
could, we should have
2a 2a
Now pi + /3i= 1 ; and therefore
il + 7^2 = pi  0, + p,  (Tj =  1,
which is impossible, as neither ky nor k^ is negative.
The condition can be satisfied for both values of a, if the
values of (7 are the same, that is, if
13 = 0:
for then the condition, that the equation {0 + l)O=j can have a
positive integer as a root, shews that the equation
„_ a^+ 7^
w  ^ w
possesses two normal integrals of the form
e^aCCo + ca+..+c^"),
e"'s(c,c,z+...±ce^'').
The condition can be satisfied for both vahies of a, if the
values of it arise through the same value of p, whether they are
the same or not ; and the equation then possesses two normal
integrals. The limitations on the constants are given in the
first of the succeeding examples.
Ew. 1. Prove that the equation
w s4 ■"■
possesses two normal integrals, if
where q is any integer, positive, u^ative, or zero, and. /i is an integer that
may not vanish. (Hamhurger.)
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91.] HAVING NORMAL INTEGEALS 281
Ex. 2. Obtain the conditions sufficient to secure that the equation
v/' + 2^vf +
[ + 3j + y«M^S^(Z*
may have a normal integral of the foregoing type. Can it have two normal
integrals ?
Eie. 3. Prove that the equation
possesses two normal integrals, if a is an integer (positive, negative, or zero).
Ex. 4. Prove that the equation
^ ■ jfl
possesses a normal integral if the quadratic equation
has a positive integer (or zero) for one of its roots for either value of ^a.
What happens (i) when both its roots are integers for the same value of ^a,
(ii) when, for each value of ^a, the equation has a positive integer for a root ?
Ex. 5. Prove that the equation
V"'2ft(n + l)^ + 4™(w + l)^4^w(™ + l)(™+3)(»i2) + <i'}«'=0,
where re is an integer and a, is any constant, has four normal integrals of the
where ^ (  ) is a polynomial in  . (Halphen.)
92. In an earlier paper, Cayley* had proceeded in a different
manner. If
where {z) is a holomorphic function of z not vanishing with z,
we have
W 3 0(S)
• CreiJs, t. c (18S7), pp. 286—295; Coll. Mn.l\. Paperf, vol. sii. pp, 444—152.
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where R(z) is a hotomorphic function of 2 in the vicinity of the
origin. Further, if
where (s) is a holomorphic fiinctiou of s not vanishing with s,
and li is a polynomial in  , we have
W S (s)
fJiw,
say, where Ii(z) is holomorphic in the vicinity of the origin.
Cayley transformed the equation by the substitution
and then proceeded to obtain, from the differential equation for y,
an expansion in ascending powers of 2. When once a significant
expression for y has been obtained, the value of w can immediately
be deduced.
Applying this method to the equation
„ ^ a + /3g + 7^ °
w ^  w,
the equation for y is at once found to be
Hamburger's investigation shews that the integrals of the equa
tion in vj are
„..,»P,(lj, „,.,»p,(l),
which are valid over the whole plane but have ^ = for an essen
tial singularity. If an integral, say w^, has an unlimited number
of zeros, the origin being its only essential singularity, then*
any circle round the origin, however small, contains an unlimited
* I', f.. g§ 32, 33.
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92.] METHOD 283
number of these zeros: so that if, in the vicinity of the origin,
the expression of w, is
^ (a) would have an unlimited number of zeros within the small
circle so drawn. The expression for ^ is
but the function ,) ! has an unlimited number of poles in the
immediate vicinity of the origin, and so the righthand side
cannot be changed into an expression of the form
where m is a finite integer. Accordingly, the assumed expansion
is not valid in this case : and the method does not lead to signifi
cant results.
But when the integral has only a limited number of zeros, so
that ^{z) is expressible in the form
in the vicinity of s = 0, where ^ f  j is a polynomial in  and /(s)
is a polynomial in a that doe;
be changed into an expansion
is a polynomial in a that does not vanish with z, then "^ can
and so the assumed expansion for p is valid in this ease. The
method therefore does then lead to a significant result*.
Assuming the method applicable, and returning to the equa
tion
* The disorimination between the cuses, and the explanation, are due to
Hambni^er, Crdle, t. cm (1888), p. 342.
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284
we easily find
SUBNORMAL
2ajao + Uj' — a, = y,
and, for any value n which is greater than 2,
2(a„<i„ + «^.ai+...} + (jJ3)a„_, = 0.
If the constants in the equation were unconditioned, the co
efficients thus determined would give a diverging series for y.
But we are assuming that the method is applicable, so that the
conditions for convergence are to be satisfied ; and then, as
.(c+a
■■),
where the last series converges. The method does not, however,
give the tests for convergence of the series for y, at least without
elaborate calculation : still less does it indicate that the con
vergence of the series for y is bound up with the polynomial
character of the series in the expression for w. It can therefore
be regarded only as a descriptive method, capable of partly
indicating the form of integral when such an integral exists :
manifestly, it is not so effective as Hamburger's.
But the method, if thus limited in utility, has the advantage
of indicating an entirely different kind of integrals of the original
differential equation, which are in fact subnormal integrals, though
it does not establish the existence of such integrals : for the latter
purpose, other processes are necessary. It will be sufficient to
consider an equation, say of the fourth order, in the form
where the origin is a pole of p^ c
Taking
multiplicity ct^, for ^ = 1,2, 3, i.
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92.] INTEGRALS
we have
— = y +^yy +y^
— = y'" + ^yy" + ^y"' + ^'y + .'/'.
so that the equation for 1/ is
if + i^f + Sy" + Gfy +y' + p, (y" + 3i/y' + f)
+ Pi iy' + y^) + Pay + P4 = 0.
If this equation is satisfied by aii expression of the form
y = z^ {a, + a,^ + ...),
the coefficient of the lowest power of z must vanish. Now the
governing exponents for the terms io succession are
TO 3, 2m 2, 2m 2,  3m  1,  4to,
— wi — m — 2, — BTi — 2m — \, — cti — 3to,
— OTs — m — I, — ^^ — 2m,
To determine which groupings of terms will give the lowest power
of z, we use a Puiseux diagram*; and in connection with each
quantity ra^ + km + 1, for the various values of fi, k, I, mark a point
{ot^ + 1, k) referred to two rectangular axes Ote, Oy. Through the
point (0, 4) take a line parallel to the axis Ox, and make it swing
in a clockwise sense until it meets one or more of the points :
round the last of the points then lying in its direction, make it
continue to swing until it meets some other point or points ; and
so on, until it passes through the point (in,, 0). A broken line
is thus obtained ; the inclination of any portion to the negative
direction of the axis Oy being tan~^/i, the quantity ^n is a possible
value of TO, and the terms giving rise to the lowest index of s in
the differential equation for y are those which correspond to the
points on that portion of the line. There are as many possible
values of m thus suggested as there are portions of the line.
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286 EXAMPLES OF [92.
It is not, however, a necessity of a Puiaeux diagram that only
integer values of m. shall thus be provided : and it does, in fact,
frequently happen that rational fractional values arise. Let such
an one be  , where r and s are prime to each other ; and take
When the independent variable is changed from s to %, an expres
sion for y <ii this type can be constructed, and it will be a formal
solution of the equation ; if the series for y converges, then such
an integral exists, expressed in the form of a series of fractional
powers, and a corresponding integral w will be deducible. Such
an integral, when it exists, is a subnormal integral.
It is easy to verify that the only points, which need be marked
in the diagram for the purpose of obtaining the possible values of
m, are those which correspond with the quantities
4m, «r, + 3m, to, + 2m, OTj + m, ra,,
as in § 86 ; but fractional values of m are now admissible in every
case, instead of being so only under conditions as in the former
use of the diagram,
Ex. 1. This indication of integrals in a aeries of fractional powers was
applied by Caylej and Hamburger, in the memoirs already cited, to the
(SS)
which possesses neither a regular integral nor a normal integi'al in
vicinity of 3=0.
The only points to be marked for the Puiseus diagram are 0, 3 ; 3,
there is one portion of line, and it gives
Accordingly, we take
and the oquation for w then becomea
^_ldw_ C^.^Cx
or, wr.tmg
w = f* W,
" This equation ie used only for purposes of illustration; its integrals
regular in the vicinity of s — <n ,
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92,] SUBNORMAL INTEGRALS
dHV
ivhich is a special fonii of the earlier equatiou in § 91. It possesses two
integrals, normal in (, if the quadratic
(l(e + l) = 4y' + f
hai> one of its roots an iiit«ger, that is, if
y = TV(25l)(25 + 3),
where 6 is any positive integer {or aero).
To tied the integrals, we have merely to adapt the solution in § 91, by
taking
a = 4p; ^ = 0, v = V + f = S(^ + l)
Thus ra = (.* = 3(3'*, o = l, and
^(ne){n + + 1)0^;
and so, taking C5 = l, we have
as a normal integral of the equation in f. Accordingly, the equation
whore e is e
I positi'
ve integer or
zero, and a ii
3 a constant, has
.
=.^i'"V
iKiiJ
' {6+n)]
.4
Manifestly,
the other integral i
s ^ven by
„
,^,u.s^
'im
4,
the two constituting a fundamental system, Eaeh of them is of the type of
normal integral ; but the aeries proceed in fractional powers of the variable.
It will bo noted that the two values of <r Bxe the same, and that only
one yalue of p is used ; the relation is
Sj:. a. Prove that the equation
where X is a constant and 2fi is an odd int^er, pc
two subnormal integrals.
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EQUATIONS HAVING
Equations op Higher Order having Normal or
Subnormal Integrals.
93. There is manifestly no reason why Hamburger's method
should he restricted to equations of the second order ; and he has
applied it to obtain the corresponding class of equations of general
order, the properties of the integrals defining the class being
(i) the integrals are of the regular type in the domain of
(ii) the origin is an essential singularity for each of the
integrals, and at le^t ono of the integrals must be of
the norma.l type in the vicinity of s = ;
(iii) all the points, except z = and z = oo, are ordinary
points of the integrals and the equation ;
(iv) the number of zeros of at least one integral, which lie
outside any small circle round the origin, is limited ;
the second and the fourth of which are not entirely independent.
Let the equation he of order n, and have its coefficients
rational. The first of this set of properties requires the equation
to be of the form
where Pups, ...,pn are holomorphic functions of s for large values
of z, and thus are expressible in series of powers of 3 of the form
a^ + h^ + G^^+ — (/i=l, ■■■,n).
The third of the above set of properties requires that every value
of z, except s = 0, shall be an ordinary point for each of the
coefficients : and hy the second of the properties, s = is a singu
larity of the equation and therefore of some of the coefficients.
Accordingly, the powersfcries for the coefficients p, which have
been taken to he rational and are limited so that every point
except 2^ = is ordinary for them, are polynomials in zr^.
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93.] NORMAL OB SUBNORMAL INTEGRALS 289
As the integrals are regular in the vicinity of s = a> , one at
least is of the form
where Q is a series of powers of z~', which does not vanish when
z=<x: and converges for all values of z outside an infinitesimal
circle round the origin, and where p is a root of the equation
p(pl)...{p^n + i) + a,pipl)...(pn + 2) + ...
the indicial equation for 2=x. The exponents to which the
integrals belong, being regular in the vicinity of z= 'x> , are the
roots of this equation with their signs changed ; and they exist in
groups or are isolated, according to the character of the roots.
Let the above integral be one which, under the second of the set
of properties, is a normal integral in the vicinity of s = 0, neces
sarily an essential singularity; in that vicinity, its expression is
of the type
where R {z) is a function of z, which is holomorphic in the vicinity
of 2 = and does not vanish when s = 0, and where fl is a poly
nomial in «~^, say
Ii =
*  1 3'«i
and (7 is a constant, Then, in the vicinity of ^ = 0, we have
VJ z^+i s™ z^ s R (z)
= r + B„
where T is a polynomial in  , constituted by U'   , and i£i ie
the holomorphic function of s given by R' {z} r R (s). But as
this arises through a form of the integral, postulated for the
vicinity of z = 0, while the integral is actually known to be
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the above form for v>'jvj must be deducible from this actual value.
This is possible only if (^[)i which has s = for an essential
singularity, possesses at the utmost a limited number of aeros
outside an infinitesimal circle round the origin ; for if it had an
unlimited number of zeros in the plane, other than ^ = 0, any
circle round the origin, however small, would include an infinite
number, and then
"<}
would be incapable of such an expansion. The requirement, that
thus arises, has been anticipated by the assignment of the fourth
among the set of properties of the integrals; and so we may
assume Q{\ to have only a limited number of zeros. Accord
ingly, as in § 91, the form of 6 {  ) must be
where P() is a polynomial in  having as its roots all the zeros
of Q ( j , and g[\ is a holomorphic function of ~ , tinite every
where except at z=^.
Let k be the number of zeros of Q ; then P fj is a polynomial
of degree k, an<l so it can be represented in the form
where {z) is a polynomial in z of degree k. Thus the integral is
of the form
The postulated form must agree with this form; hence (I [\
is the polynomial li of that form, and the holomorphic function
R{p) of that form is the polynomial G{z) : also
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93.] INTEGRALS 291
The expression for w'/w in ascending powers of z is thus valid,
under the conditions assigned, provided ii(s) is a polynomial in z.
Taking
so that Pi is a function of z, which is holomorphic in the vicinity
of 3 = and is equal to a™ when 2 = 0, we have
Then
10 \w) dz\w}
= Z^^ {P,= + 2^Qx) = ^''"T'P,,
say. Similarly,
"w = ^~"''" ^ ^'' '•' ^"' ^'* " ^"""""Ps,
say, and so on: where all the functions Pg, P3, ..., Qj, Qj, ... are
holomorphic functions of z, and the first m terms in P, ai'ise from
Pi". Substituting in the equation
d'^w d'^^w dw
^2" ^ d3"~^ ' dz '
we have
P„ + z'^ih Pnl + 2'>=P»s + ■ . ■ + 2"™p«  0,
which must be identically satisfied. The coefficients p are poly
nomials in : hence*
z
z^p,
is expressible as a polynomial in z, and so the highest negative
power in p^ is z~'™ at the utmost. Accordingly, let
J ^ z z^ z'^
Now we have
F^ = rt,„\ «™_,2 + ... + MiS""' + ^2"= v + ^2",
•^ If this were not the case, the aesignment of a larger value of m could Bet
it: SDd GO the assumption reailj is no limitation beyond that wliicli is neoesBarj
a noiTEial integral, viz. la must be a finite integer.
19—2
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292 CONSTRUCTION OF [93.
say, where T is a holomorphic function of z ; and
where T^x is a holomorphic function of s ; so that the first m
terms in P,, which give all the coefficients in the exponent of the
determining factor e", are given as the first in terms of a root of
the equation
u" + z™pi ii"~' + s™^5 !!"= + . . . + Z^^'pn = 0,
when the root is expanded in ascending powers of z. When the
first «i terms in v are obtained, then the determining fa<;tor is
known ; for we have
li= {' ar"^^v
Moreover, after this determination, the terms involving the powers
^, z\ ..., ^"^' in
have disappeared, so that this quantity is divisible by z™, leaving
a holomorphic function of s as the quotient.
94. Having obtained the determining factor, let
be substituted in the differential equation, which can now be
taken in the form
For this purpose, derivatives of e" are required. We have
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94.] NORMAL INTEGRALS 293
and so on, where v is identical with the first m terms of Pi,
Vi is identical with the iirst m terms of P^, and generally, Vx is
identical with the first m terms of Pk±^. Now
^ _ e" 2 I  J^ji^^"^
with the convention v„ = v, Ui = 1 ; and therefore the equation
for u, after dropping the factor e^, is
which can be written in the form
where j)o= 1 The coefficient of m is
Because the first to terms in ^ai are the same as in P^, the first
TO terras in the preceding coefficient are the same as in
P™ + ^'"PiPni + . . . + ^""Pn,
and they are known to vanish, for the coefficients of s", s'', ..., 2"'~'
were made zero to determine v ; hence the preceding coefficient is
divisible by a™, so that we can take
if„_, + £"'piV„_^ + ... + z'"^p^  2^(0, + e,z + ...),
where 6^ is a determinate constant, because v is known.
The coefficient of 2™+^ ^ is
dz
= mUna + (Jl  1) IJ,^3™^1 + . . . + Sv^f" "i)^a + ^f"^^' "'p^i.
The first m terms here are the same as the first to terms in
mP„_i + {n l)z"'p,P^, + ... + aPis'^^'^^^H af^^iJ^i,
that is, the same as the first m terms in
„„«! +(n~l) v'^^z'^p, + . . . + 2vz("^^ "'p^^ + z '"" ^'pnv
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294 CONDITIONS FOR EXISTENCE [94.
The equation for v is
H" + 3™Pj^"' + z^"'p^v'"^ + ... + z"™p„ = ;
and, in particufar, the equation determining a^, the constant term
giving n values of o^,
95. Let a,„ denote a simple root of this equation, sometimes
called the characteristic equation : then th^ quantity
«£»«,''"' + (»~l)«™""'ni,» 4 ... + «„_i,«mm
is not zero. The coefficient therefore of z™^' y , as given above,
does not vanish when s = r let it be
'7o + '?i2+ ■.
where 5jo is a determinate constant, because v is known.
It follows that the equation for it, in the form as obtained, is
divisible throughout by s*^. Further, if it possesses (as, for the
class of equations under consideration, it must possess) a regular
integral, and if that regular integral belongs to the exponent a,
then a is given by the' indieial equation
V„<y + t^o = 0,
so that u can now he regarded as a known constant.
Further, we had
where & is a positive integer (or zero), and p is a root of the
equation
p{pl)...{pn+X)^p(p\)...{pn^2)a,,
+ p{p\)...{pn\ 3)aai + ... + /5a„_i,o+ano=0,
say, of
'(p) = 0.
Consequently, the equation
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95,] OF NORMAL INTEGRALS 296
regarded as an equation in k, must have at least one root equal to a
positive integer or zero : if this root be denoted by k, one condition
that u should be of the form
u = a" {c„ + c^e + . .. + c^if)
(which is the fonn for u required by the earlier argument) is
But while the condition is necessary, it is not sufficient for the
purpose. When the value of u is substituted in the equation, the
latter must be identically satisfied ; and so we have relations
among the coefficients c. The general relation is
/(o + a) o.+g,(a)c^^,+g,(a)c^, + ... + £r™„^™(a) c,+™^™ = ;
the relations for the first few coefficients are of a simpler form.
When these relations are solved, so as to give successively
the ratios of c,, Cj, ... to Co, a formal expression for u is obtained.
In this forma! expression, all the coefficients c,+,, c^+2, ... are
to vanish ; that this may be the case, we must (as in § 79) have
/(ff + «)c, = 0,
7 (<r + «  1) c,_i + 9, («  1) c,  0,
/(<7 + «2)c,_ + ^,(«2)c,_, + 5.(«2)c, = a
and so on, being m.(n — 1) relations in all. Of these, the first is
known to be satisfied as above; it is the first condition for the
existence of u in the specified form. The aggregate of conditions
is sufficient, as well as necessary: the last of them secures that
c,+i vanishes, the last but one secures that c^+j vanishes, and so
on : the first secures that c»+mnm vanishes ; and then, in virtue
of the general differencerelation among the constants c, every
succeeding coefficient vanishes.
Thus when the m (ji — 1) conditions arc satisfied, in association
with a simple root of the equation
a normal integral of the original equation exists.
It may happen that the conditions are satisfied for more than
one of the simple roots of the equation : then there will be a
corresponding number of normal integrals of the equation.
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296 NORMAL [95.
The extreme case would be that in which every root of the
equation
is simple and the conditions are satisfied for each of the roots ;
there then would be n normal integrals. Let the n roots be
denoted hy 6^, 8^, ..., On, so that, if
the normal integrals will be of the form
where Uy is a polynomial, say of degree Xr, in s We have
(Tt = p  Kt ; when these n indices a,, arc associated with n
quantities p, it follows that
for r = l, ...,». The distinct quantities p^ are the roots of
/(p) = 0, so that, if they are all different from one another, we
have n of them ; also
i_(„, + ,,) = i„(,>l)a,..
The value of 2 o^ can then be obtained as follows.
Construct the fundamental determinant
\ dz ' dz ' ■"' dz
which is equal to
that is, to
rft.
where A 1
1 constant. Now if w^ = 1 + c„j
yGoosle
95.] INTE(3RA.LS 297
where u^, is a polynomial in z which is equal to 1 when s = ; also
where Un is a polynomial in s which is equal to 1 when e = ;
and so on. Thus
„+^„+..i.,.
)(.+i)(i.(^),
2).
1 +...
1 +
e, +...
9, +
«,■ + ...
«,"+,.., ...
As the roots are unequal to one another, <I>(z) does not vanish
when 3 = 0; and it is a polynomial. We thus have
*"l'''"'"+"3><ii)^«~(
Accordingly
iii + ... +fl„ = — + ... +
1 Ci.ir
^S<r,i»(»~l)(m + l) = «,,
that is, "l" (s) reduces to its constant nonvanishing terra. Thus
i^„,i»(»l)(m+l)o,..
We saw that
I (,r, + «,) = i»(nl)«,.;
which is impossihJe because no one of the integei's k, is negative.
It therefore follows that when the characteristic equation
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298 MULTIPLE ROOTS OF [95.
has all its roots distinct from one another, and when the quantity/
denoted by u has n distinct values, associated respectively with
n distinct roots of I(p) = 0, the differential equation
where
cannot have more than n — 1 normal integrals, linearly independent
of one another.
If, however, the quantity denoted by a liaa fewer than n
distinct values, so that it could he the same for more than one
of the n distinct quantities li, the relation
l^^. = i«(«l)(m+l)a,„
would still hold, repetitions occurring on the lefthand side. But
in that case not all the roots of the equation / (p) = are specified,
for the same value of le could be associated with the value of
a common to two integrals; and the relation
S(,T + «) = i«(»l)a..
no longer holds. The theorem then cannot be inferred as neces
sarily true : and it will appear from examples that an equation in
such a case can have a number of normal integrals equal to its
order.
Similarly, if o has n distinct values, and if these values are
not associated with n distinct roots of / (p) = 0, the preceding
theorem is not necessarily true; the differential equation can
have a number of normal integrals equal to its order.
96. Next, let a„ denote a multiple root of the characteristic
equation
a™" + Om""' ffli, im + ■ . ■ + c[»,«™ = ;
then the quantity ?jo vanishes, where
lJ, = )iam"^ + {>l — l)a,^"'^«i,im + ... +ani,wmm
The indieial equation is
and cr must be a finite quantity. If 0„ is not zero, the latter
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96.] THE CHARACTERISTIC EQUATION 299
condition is not satisfied: and tlien the original equation lias no
normal integral to be associated with that multiple root. If B^ is
zero, the preceding indicial equation is evanescent : and so further
consideration is required. The differential equation for «, on
division by 3™+', becomes
dn
(»,+
9,J+.
+ 2+> (*. + *,
wh.
ere the
coefficient of ',^ is of the form
for
r = 3,
4
2'+'"'~"(t. + +.^ +
When
ni^l,
the indicia] eqnation is
when m >
l.the
indicial equation is
In either case, we can have a possible value for cr. A regular
integral of the equation for ti, and a consequent normal integral
of the original equation, exist if the appropriate conditions,
corresponding to those for a simple root, are satisfied : it is
manifest that they become complicated in their expression*.
97. It might happen that, in determining v, one or more
roots of the equation
«7«" + «!»""' «!, im + ■ ■ ■ ■+ (^i, nm —
is zero, while some of the remaining coefficients in v do not
vanish ; the implication is that (other conditions being satislied)
a normal integral exists, having a determining factor of which
the exponent is a polynomial with a number of terms less than m.
It might even happen that, with a zero value of a^, all the associ
able values of the rest of the coefficients are zero, so that v = 0, and
the determining factor disappears. One possibility is the existence
of a regular integral, and the possibility can be settled in the
particular case by the method given in Ch. vi. If, however, the
conditions for a regular integral are not satisfied, then there is the
' They are considered by Giinther, Cnlle, t. cv (1889), pp. 1—34, in particular,
pp. lOetseq.
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SUBNORMAL [97.
Hty of a subnormal integral of the original equation ; it
IS follows.
Lot
«.=(«,.
substituted in the equation
d'Hv
"dz^ ^
!£!
fj»=o;
en the
equation for \
.,. is (by § 85)
S+fi
^)S
... + g.«.
■0,
lere
d
?,
f,.
Now fi is to be chosen so as to diminish the multiplicity of a =
as a pole of g„. After the preceding hypotheses, we shall not
expect to have an expression of the form
(1/ _ Uj , ^ I , _^~''i__
where m is an integer; but after the indications in § 92, it is
possible that II' may be a series of fractional powers. Accord
ingly, assume that the multiplicity of s = as an infinity of li' is
/t, so that ^''fi' is finite when 3 = 0: then in ^n, we have a series
of terms with infinities of oiders
n/i , (n l)/i 1 , ...
(jil)^+ m + l, («~2)^ + m + 2, ...
(?i. 2)^+ 2m + 2, (n3)y^ + m3, ...
n{m + \).
Construct a Puiseux tableau by marking points, referred to two
axes, and having coordinates
0, « ; \, n\\ ...
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97.] INTEGRALS 301
(it is easily seen to be necessary to mark only the first in each
row), and construct the broken line for the tableau, as in § 92.
If the inclination to the negative direction of the axis of y of any
portion of the line is tan"' 0, then ^ is a possible value for ft. If 6
be a positive integer ^ 2, we have a case which has already been
dealt with. If ^ = 1, there may be a corresponding integral ; but it
is regular, not normal. If ^ be a negative integer, li' is not infinite
for 3 = 0, and the value is to be neglected. If ^ be a positive
quantity but not an integer, it must be greater than unity to be
effective ; for if it were less than unity, H would not be infinite
for s = Q. Suppose, then, that 6 has a value greater than unity;
as it arises out of the Puiseux diagram, it must be commen
surable : when in its lowest terms, let it be
where q and p are integers prime to one another, and q>p.
Then take
3 = a* ;
we have an equation in u and x, and a possible determining factor
e" can be found such that
and so
a series of fractional powers. The investigation of the integral of
the new equation in u and x, that may exist in connection with
this quantity fi, is of the same character as the earlier investi
gations.
Equations of the Third Order with Normal or
Subnormal Integrals.
98. The preceding general theory, and the methods of >
with the cases when the equation for a^ has equal roots, or h
zero roots, may be illustrated by the considei'ation of an equati<
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302 EQUATIONS OF [98.
of the third order more clearly than by that of an cqua.tion of the
second order, as in § 91. Taking the ainiplest value of m, which
is unity, the equation is of the form
", ^^lo^ + ^ji '/ k«,sf + kill ^ k,« ,
w +3 , — w H — — — ; w
fC^Z j iC^S + jCj^Z + fi^aa r. ,
which, on using the substitution
ti,„z+k„ ^^ iu
y = we ^ =iuz''">e ',
becomes
where the constants a are simple combinationB of the constants k.
The substitution adopted changes a normal integral of the one
equation into a normal integral of the other, save for the very
special case when it might be changed into a regular integral of
the other: it therefore will be sufficient to discuss the form
which is devoid of a term in y".
In the present case, m = 1, we take
y = e'u,
and a is chosen so as to make the coefficient of the lowest power
in the coefficient of w equal to zero. We thus have
and the equation for u then is
+ , [a^e'' + («si  a«™  6a) a + (a,,  ««,.  6^0) = 0,
of whicli the indicial equation for s :^ is
It is clear that the equation in a. will not have a triple root : if it
could, we should have (Xs = 0, «3a = 0, a = 0, the last of which
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98.] THE THIBD OBDER 303
values leads to the collapse of the process. (Account must, of
course, be taken of the possibility that csja = = a,^, and this will
be done later.) Meanwhile, we assume that a is either a simple
root or a double root.
First, let a be a simple root ; then o^a + 3a^ is not zero, and
the foregoing indicial equation then gives a proper value for a.
If — p ia the exponent to which an integral in the vicinity of
0—x> belongs, p is a root of the equation
/{p) = p(pl)(p2) + a„p + «,, = 0.
The general investigation has shewn that this must have a root of
the form p = o + k, where k is a positive integer (or zero), and
that, if this condition is satisfied, the form of u is
M = 3''(C„ + C,S+ ... +0,3").
We substitute this value, and compare coefficients. If
ff„ = {ff + n){ff + nl)(<7 + ii2) + a^(a + n) + a^,
k„ = SaX<T+n)(a + n + l)+{a,, + 6o.){^ + n + l)
+ Ug — aoan — 6a,
then the difference equation for the coefficients c is
for values of m^O, together with
As a is a simple root of its equation, a^, + 3a^ is not zero : thus all
the quantities k^,, k^, kj, ... are different from zero, and the pre
ceding equations thus determine Ci, C3, ... in succession, say in the
form
In order that the integral may not become illusory, the series is to
be a terminating series : it would otherwise diverge, on account of
the form of §„■ Let the series contain k + 1 terms ; then all the
coefficients c»+i, c,+a, ... must vanish. Now c,+i vanishes if
then c,+a vanishes if
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304 NORMAL INTEGRALS OF AN [98.
and then all the succeeding coefficients c vanish. The latter
condition gives ^,= which, as ^„=/((r+«,) for all values .>■'
n, is the same as
/(„ + «).0,
a known condition ; and the other gives
which is the new condition. When hoth conditions are satisfied,
a noiTTial integral exists for the equation in y. As that equati i ■
involves seven constants, which are thus subject to two conditio:!.,
there are effectively live constants left arbitrary, subject solely :■■ 
a condition of inequality as regards the roots of the equation
moreover, « may be any positive integer (or zero).
If the corresponding conditions hold for a second simple root
of this cubic equation, the number of independent constants is
reduced to three, while there are two integers such as k; the
differential equation for y then has two normal integrals.
If all the roots of the cubic equation are simple, and the
corresponding conditions hold for each of them, there are three
integers such as k, and there is effectively one arbitrary constant :
the differential equation for 1/ would then have three noi'mal
integrals. This, however, is impossible, if there aie three diff<!renfc
values <r, a', <r" of a, and three associated integers k, k, k", yiich
that (7 4 K, <t' + ic', <!■" 4 k" are different roots of f{p) = 0. For
then
Now we have
loo? + attji  ^35
where
dk
da
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98.] EQUATION OF THE THIRD ORDER
hence, summing for the three roots of h, we have
by a well known theorem in the theory of equations. We then
should have the equation
which is impossible as no one of the integera k, k', k" can be
negative. Hence, when the equation a? + tto^B — Osj = has three
distinct roots, and when there are three different values a, a, a"
of IT, associated with three integers k, k, k", such that <r + «, (r'+ k,
o"+ k" are different roots o{f(p) = 0, then the differential equation
cannot have more than two normal integrals. But, if the values of
<j are fewer than three in number, or if the quantities a + k are
not different from one another, then the differential equation (the
other conditions being satisfied) can have three normal integrals.
Next, let a be a double root of the equation
ft. = a* + aiXffl — asa = 0,
BO that we have
in order that this may be the case, the relation
must be satisfied. The quantity a, given by
(3a^ + £%) (7 + o^y ~ aos, — 6a^ = 0,
is infinite, unless a^ — aaj, — 6tt' vanishes : if this condition is not
satisfied, then the regular integral for the wequation, and conse
quently the associated normal integral for the yequation, cannot
exist. Hence a further condition for the existence of the normal
integral is, that the equation
Ksi — noai — 6a^ =
be satisfied, where a is the double root ; that is,
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306 EQUATION OF THE
Assuming this to be satisfied, the equation for u aow is
Now
Sctj,
21, say;
so that the equation for u is
The indicial equation for z — is
Substituting
« = 3«(p„+C,3+... +C„3''+.,.)
in the equation, we have
Du = Cc^S {C,,f (fl  1) + C.,ye + Osij,
provided
for all values of )i ^ 0, where
g^=^{e^n)(e^n^l){8 + n2) + a^{e+n) + a^,
A„ = c,„(S + m + ])(^ + m) + C^(^ + ?t + l) + c„.
First, let the roots of the indicial equation be unequal, say X
and /i, so that
Du=c,o,„e^{e\){e (>.).
Then the value of u, when 8 — X, gives an expression which
formally satisfies the equation ; but it has no functional significance
unless the series converges. That this may happen, g^ must vanish
for some value of n, say k^, when d ~X; that is, one root of
I(p) = p(pl)(p~2) + a^p + a,, =
must be
pT=\+ Ki,
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98.] THIRD OllDER 307
where «i is a positive integer or zero. If that condition is
satisfied, then a regular integral of the wequation and an asso
ciated normal integral of the ^/equation exist.
Similarly, if l{p)~0 has another root
p = fi + «:.„
where «, is a positive integer, then the value of u, when $ = /j., has
significance. It is a regular integral of the wequation; and a
corresponding normal integral of the original equation then exists.
Let denote the root of the cubic that is simple : then the
earlier investigation shews that a corresponding normal integral
may exist. If a' be the exponent to which the regular wintegrai
belongs and H k^ + I he the number of terms it contains, then the
equation /(p) = has a root
p = a' + >c,.
But the three normal integrals, each one of which is possible,
cannot coexist, if X + ati, /j, + k^, ct' + k, are different roots of
/(p) = 0, supposed not to h&ve equal roots. If they could, we
should have
>. + /i + (7 j Ki+ IC^ + Ks — ^p — S.
Now
X4w = l — = 1  — .
Also
and
for a, a, are the roots of the equation
a^ + aa^i — agj = ;
on reduction, after using the value of a and the relation
fflisa — "«ai — 6a° = 0.
Hence
\ + fi + <r'^5,
and therefore
/C i /Cs + «3 = ~ 2,
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308 NORMAL INTEGEAL8 OF AN [98.
which is impossible, as no one of the integers « can be negative.
Hence, when the roots of the indieial equation
C„.r(,rl) + C„.r + «. =
are unequal, acd wheo /(p) = has not equal roots, the original
equation cannot have more than two normal integrals, unless (in
the preceding notation) there are equalities among the quantities
X + Ki, /i+Ka, cr'+zcj. If it possesses the two normal integrals
associated with X and fi, it is easy to see, from the expression for
kn, that, if \ — /j. be a positive integer, it must be greater than
Ka + 1 : and that, if /t — X. be a positive integer, it must be greater
than «i + 1.
Next, let each of the roots of the indieial equation for <r be
equal to r : so that
Thus the two quantities
."(^t)^.
M [!],..•
are expressions that formally satisfy the equation : they have no
significance unless the series converge. That this may happen, g^
must vanish for some value of n, say k, when d — r; that is, one
root of the equation
I(p) = p{p 1) (p  2) + a,„p + «,„ =
must be p — t + k,
where «' is a positive integer or zero. (The quantity A„ never
vanishes in this case and so imposes no condition.) On dropping
the coefficient Co, the expression for u in general is equal to
■''/i„A/' ■■■^* ^^AA.
so that the two integrals are of the form
V, vlog^ + iJi,
where v — [u]e=T, and Vi is an expression similar to v with different
numerical coefficients, viz. the coefficient of (— lyz^'^'' in v, is
[hX .*,, Ui \g. ~tie K dei]\,„ ■
The corresponding normal integrals are
<fv, «■(» log « + ..).
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98.] EQUATION OF THE THIRD ORDER 309
A third normal integral can coexist with these two in the present
case in the form
where u belongs to the exponent c', = — + 4, provided l{p) =
has a root of the form a' + x^, where «, is a positive integer (or
zero). The reason why three can coexist in this case is that
only two quantities t and (/ arise, and only two roots, not three
roots, of I (p) =0 are assigned.
B^\ 1. Prove that, if the equation
possesses a, normal integral of tlie form
the constants j3, n, o are given by the relations
(T (3S= + oao) + 3a=S  93^ 4 ("Is, + /3aj2 + ctaa = ;
and the equation
p{pl)(p3) + p«j4 + <%'=0
must have one root equal to tr+K, where ic is a positive integer (or zero).
Obtain the relations sufficient to secure that the series Cg+OjB+.,. shall
contain only k + 1 terms.
Assuming that three values of a, distinct from one another, correspond to
three seta of values of a and ft prove that their sum is 9 : and hence shew
that, in this case, the differential equation cannot hive more than two normal
In what circumstences can th i fl t 1 i n i tl 1
integrals ?
En). 2. Obtain the constant and tl e nd t ns of e ten e f the
normal integrals of the equation n the p e ed ng e a nple when a an she?
and a^ does not vanish. How m ny n mal tgl ntlequtn then
have?
99, We now have to c d ( ) tl e ca n 11 ne ze o
root for a occurs, so that a^ — 0; and (ii), the case in which all the
roots a are zero, so that a^ = 0,0.^^ — 0.
Taking a^ = 0, the equation is
/ + zi — 2/ + —3 — — y=u.
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310 srEoiAL [99.
Two nonzero roots arc given by
a normal integral may exist in connection with each of them.
The indicial equation for « = is
in connection with this exponent, a regular integral may exist.
The investigation of the respective conditions is similar to pre
ceding investigations.
Now substitute in the equation
the equation for u is
u"'+3u"n' + u'Uil''' + il"i
a^± a^iZ + Oi^X _ _
and by proper choice of il, the multiplicity of a = as a pole of u
is to be diminished. Assume that z~i'Df is finite {but not zero)
when z = 0, and form the tableau of points in a Puiseux diagram
corresponding to
3U, 2/4 + 1, iM + t, /1 4 4, 5,
that is, insert the points
0,3; 1,2; 2, 1 ; 4,1; 5,0.
The broken line consists of two portions : one of them gives /* = 2,
the other gives fi = \. The former gives the possibility of two
normal integrals : the latter gives the possibility of one regular
integral as above.
But now let a23 = 0, as well as Us^ = 0. The equation for a
becomes
a= = 0,
so that the method gives no normal integral. When we proceed
to the equation for u, the coefficient of it is
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99.] CASES 311
We form the tabloau of points in a Piiiseux diagram corre
sponding to
d/j., V + 1. f^+^' /' + ''. ^:
that is, we insert the points
0,3; 1,2; 2, 1 ; 3,1; 5,0.
There is only a single portion of line ; it gives
/i = f.
Accordingly, we change the independent variable by the relation
s = a^;
the form of il' is
of «*
that is,
dll_3a' W'_0 ^
rfa; ~ ** "•" a^ "a^'^x''
say. The differential equation
with the substitution y^vx', becomes
+ ^ {t1a.^ + {tla,, + 180=,) ^ + {^la^ + 18<i™ + 8) a^j = 0.
If a determining factor exists, then (Ex. 1, § 98) it is of the form
^ + 2?o.. = 0,
that is.
/3
a. 3^= + 9(12,3 = 0,
 (tjittg
Substituting
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[99.
and using these values of a and /3, we find the equation for u i
the form
dx'
+ " [ 9;8= + {o? + 63aa + 27a,0 x + (12a,„ + 4) /3  6a^] a^^
+ a (9a„  2) it^ + (27tt™ + 18a» + 8) a^] = 0.
If the equation in ^ is to have a normal integral, this equation in
u must have a regular integral belonging to an exponent cr, where
it is easy to see that
The regular integral for u is of the form
u = X c,,a:^+\
If
/„ = {n + 3) (« + 2) (n + l) + (9a™ S){n + 'S) + 21a,, + lSc(,„ + S,
g^ = 3a(H + 4) {n + 3)  6a (« + 4) + « (9<i»  2),
A„ = 3/3 (■« + 5) (« + 4) + (3a^  9,9) (n + 5) + {\2a.^ + 4) ^  6a^
& = a=+63a.s, + 27aai,
i„ = 3^^(^ + 4),
the differencerelation for the coefficients c is
together with
fe(, + LiCi = 0,
AaCfl + tei + ZjC; = 0,
^_iC„ + A_iCi + ACa + Li^s = 0,
The conditions, necessary and sufficient to ensure that the aeries
for M terminates with (say) the (« + l)th term, which is the
generally effective manner of securing the convergence of the
series, k being some positive integer or zero, are
four conditions in all.
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99.] CASES
Assuming these satis6ed, we have
M^/i 2 (
a subnormal integral.
If the conditions are satisfied for more than one of the cube
roots of djg, then there is more than one integral of subnormal type.
Moreover, the value of ct is the same for all three cube roots, and
only one value of « is required : so there may be even three sub
normal integrals, each containing the same number of fractional
powers.
In order that this analysis may lead to effective results, it is
manifest that «3a should not vanish.
Hx. 1, Prove that the equation
possesses three subnormal integrals.
Ex. 2. Discuss the integrals of the equation
Normal Integrals of Equations with Rationai.
Coefficients.
100. In the discussion at the beginning of this chapter, the
only requirement exacted from the coefficients was as regards
their character in the vicinity of the singularity considered: and
a special limitation was imposed upon them, so as to constitute
Hamburger's class of equations in §§ 91—99. More generally,
we may take those equations in which the coefficients are rational
functions of z, not so restricted that the equations shall be of
Fuchsian type ; we then have
Po jiiT ^Pi JZazT + ■■■ +P»w = 0,
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314 roiNCARfi ON [100.
where pa, P\, ■■■, J>n s^rs polynomials in s, of degrees to,,, ot,, .... w,;
respectively. The singularities of the equation are, of course, the
roots of po = and possibly ^ = oo ; owing to the form of all
the other coefficients, it is natural to consider* the integrals
for lai^e values of \z\.
It will be assumed that the integrals are not regular in the
vicinity of z = oo . When a normal integral exists in that vicinity,
it is of the form
where is a uniform function of z~' that does not vanish when
s = x , aJid O is a polynomial in z of degree (say) to, so that the
integral can be regarded as of grade m. As in ^ 85—87, the
value of ii' is obtained, by making the m highest powers in the
expression
i)oa'" + p,0'"' + ... +p„
acquire vanishing coefficients; and a Puiseux diagram at once
indicates whether a quantity fl' of such an order can be con
structed. The value of m — 1 is the greatest among the
magnitudes
provided two at least of them have that greatest value, which may
be denoted by k. Then for such normal integrals as exist, we have
when k is an integer, and
where [h] is the integral part of h, when h is not an integer. The
integrals are of grade ^h + 1, or ^ [h] + 1, in the respective cases ;
and the equation is of rank k + 1.
Take the simplest general case, when the equation is of rank
unity, and when, in the vicinity of 3= co , it may possess n normal
integrals which, accordingly, must be of grade unity. No one of
the polynomials^, ...,pa is of degree higher than p„; assume the
degree o?po to be k, and let
* See Poincar^, Jnier. Joiirii. Math., t. vii (1885), pp. 30.^—268; Acta Math.,
t. vjij (188(5), pp. 295—344.
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100.] NORMAL INTEGRALS 315
where some (but not all) of the coofficients a may be zero and, in
particular, where it will be assumed that a^ and «„ differ from zero.
The determining factor for any normal integral is of the form e^' :
6 satisfies the equation
fr,(f) = a„^"la.(9'' + ...+fl„_.^ + a„ = 0.
The preceding theory then shews that, if the roots of this equation
are unequal and are denoted by 8^, 9^, .... 6^, the normal integrals
are of the form
the quantities cr^ are given by the equations
where
and ^i,(jh,, ^nare uniform functions of r"', which do not vanish
or become infinite when s = x . Special relations among coefficients
are necessary in order to secui'e the conveigence of the infinite
series <p ; unless these conditions are satisfied, the foregoing
expressions only formally satisfy the differential equation and,
as integrals, they are illusory.
Ei^. 1. Prove that the equation
possesses three normal int^rals in the vicinity of j; = co , when a is a positive
integer not divisible by 3 ; and obtain them.
.Kk. 2. Prove that the equation
pOBseisses three subnormal int^rals in the vicinity of x= re , when
(9'
m being an integer not divisible by 3 ; and obtain them. (Halplien.)
.Ec, 3, Skew that the equation
has two normal integrals in the vicinity of 3^='xi ; and, by obtaining them,
verify that the points a;= 1, x^ ~i are only apparent singularities.
(HaJphen.)
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EXAMPLES [100.
Show that the equation
one integral, which is a polynomial in x, and two other integrals,
normal in the vicinity of a; = to . (Halphen.)
Ex. 5. Prove that, if normal integrals exist for the equation
the constant a must be the product of two consecutive integers. (Halphen,)
Sx. 6. Prove that, if all the singularities for finite values of 2 which are
by tbe integrals of the equation
are poles, and ii pt,Pi, ..., ^„be polynomials ins such that the degree of p^
is not leas than the greatest among the degrees of pi, ,.., p„, then the
primitive of the equation can be obtained in the form
where the constants qj, ..., q„ are determinate, and all the functions i^i, ...,
<^ are rational fiinctions of s. (Halphen.)
Ex. 7. Apply the preceding theorem in Ex. 6 to obtain the primitive of
the equation
where n is an integer ; also the primitive of the equation
(ii) „+':?'rf('?'+o)„.o,
where ii is an integer prime to 3. (Halphen.)
fii'. S. Similarly obtain the primitive of the equation
aT5(a^l)y'3^e(^+.»:i'l)y=0,
in the form
(MatK Trip., Part ii, 1895.)
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101.] Laplace's definite integral 317
PoiNGARfi's Development of Laplace's DefiniteIntegral
Solution.
101. Several instances, both general and particular, have
occurred in the preceding investigations in which formal solutions,
expressed as powerseries, have been obtained for linear differen
tial equations and have been rejected because the powerseries
diverged. These instances have occurred, either directly, in
association with an original equation, or indirectly, in association
with a subsidiary equation, when an attempt was made to obtain
regular integrals of an equation, some at least of whose integrals
were not regular; and they have arisen when an attempt has
been made to obtain normal integrals of an equation, which is of
the requisite form but the coefiicienta of which do not satisfy the
latent appropriate conditions.
In such instances, the expressions obtained for formal solutions
do not possess functional significance. But Poincar^ has shewn
that it is possible to assign a different kind of significance to such
solutions in a number of cases. In particular, there is a theorem*,
due to Laplace, according to which a solution of the given diifer
ential equation with rational coefficients can be obtained in the
form of a definite integral; this solution has been associatedf by
Poincare with the preceding results in § 100 relating to normal
integrals. For this purpose, let
where the contour of the integral (taken to be independent of e)
will subsequently be settled, and T is a function of t the form of
which is to be obtained. If this is to be a solution of our equation,
we must have
or, if
r7, = 6,(»)6,t«'f... + 6„
* See my Treatise on Differential Equahong § 140.
+ In the memoirs ijuoltd in the fdolnote un p 314. The following exposition
s based paitly upon tlieie memciri, paiOv npon Picard's Ctiars d'Jnalyse, t. ill,
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318 Laplace's [101.
the necessary condition is
({U^z" + Ujz"' + ... + Uk) e''l'dt = 0.
Let
F,."TO.^.'.(TO._,)+...+(l)'a('''^.).
forr=l, 2, ...,h. Then
for each of the h values of r ; and the value of [e*^ F",] depends
upon the contour of the definite integral. Using this result, the
above condition becomes
[le"r,]+/.»{TO(m.) + ... + (i)>.(m)}.i< = o,
which will be satisfied, if T bo a solution of the equation
and if the contour of the integral be such that
The equation for T is
„ d'r /, ill, „\ di'T
so that its singularities are the roots of (/„ = 0, that is, are the
points d,, Os, ■■; ^n, and possibly infinity. Writing the equation
in the form
the value of Pj when t is infinite is 
on. Further, the quantity S Vy involves derivatives of T up to
order kl inclusive.
This equation for T has its integrals rcgnlar in the vicinity of
each of its singularities 6^, 6^, ..., $„: their actual form will he
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lOJ.] DEFINITE INTEGRAL 319
considereil later. Let "^^ denote the most general integral of the
equation for T in the vicinity of Sg; then, assuming that the
conditions connected with the limits of the definite integral can
be satisfied, we have an integral of the original differential
equation in the form
^je''^,dt,
and this result is true for s= 1, 2, ..., n. Now '^g is certainly
significant, because it is a linear combination of & regular integrals
of the equation for T; hence we have a system of w integi'als of
the original differential equation.
102. This system of k significant integrals can be transformed
into the system of n normal integrals, when the latter exist.
They can be associated with the formal expression of the n
normal integrals, when the latter are illusory.
A preliminary proposition, relating to the given differential
equation, must first be established*. In the first place, let it be
assumed that all the constants in the equation for T are real, and
that T and t ai'e restricted to real values. That equation can be
replaced by the system
dt "
When we substitute
r,. = 0^e*', (r=0, 1, ...,kl),
with the conventions that T,, — T and %„ = 0, the modified system
^^ =  P^Q  P^^e,  ...  P,0j._.  (P,  X) @fc.
' It is due to Llapounoff (lfi92i see Hoaid, Cours d'Anabju,X. ill. p, 36:
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320 liapounoff's [102.
Hence
i^(@'^ + ««),= + ... + 0v,)?^(®=+ ©/+..+ ©Wj+C^POeVi
+ ©Hi + ©.e, + . .. + «H)s_@i_i
 Pft 00*^,  ...  p,ei_,0fc_3 .
Take a real quantity (o, smaller than the least real root of [/"„ = ;
as ( ranges along the axis of real quantities between — co and t^,
all the quantities Pj, Pa, ..., P^ remain finite. Hence, by taking
a sufficiently large value of \, the quadratic fonn on the right
hand side can be made positive for that range of values of t ; and
therefore, as t increases from — «) to i„, the quantity
steadily increases in value. Consequently, when ( decreases from
(j to — =o , the quantity
steadily decreases in value. As („ is not a singularity of the
equation, the values of 0, @i, .... 0j_i for any integral that exists
at % are finite there ; their initial values are finite, and therefore
each of the quantities [0], j@,, ..., 0v_i remains finite and
decreases steadily, as ( decreases from 4 to — o^ . Hence the
quantities
all remain finite within that range, that is, no one of them can
become infinite, for a value of X sufiiciently large* to make the
quadratic form positive.
Next, suppose that the constants are complex, so that T, Ti, ...
can have complex values ; but let t still be real. Then we write
's Differential Caieulus, Srd ed., p. 408. In the
so that it is auiKeient to take \ greater thiin the greatest positive value which
malies the leithand side in the last inequality vauisli.
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102.] THEOREM 321
for all values of r, where 0^ and ir^ are real ; the system of
equations takes the form
t=*™. t=t.« (.=0,1 ,,M
Oa 0i_2 + qi ■^i..i — ... — ^10 + q,,^
where 0o=0, '^„ = ylr, and Ps=Ps + iqs We now have
equations; on substituting
they give the modified set
 9^ •J'  j)j,^
= \ 2 (4>r' + ^r') + (X  Pi) (<['%i + ^t"*l) + bilinear terms.
As before, by choosing a sufficiently large value of X, the right
hand side can be made always positive. Then, by taking a value
io smaller than the least real root of ?7o = 0, and by making t
decrease from ft, to — m , so that all the quantities p and g are
finite, it follows that, for such a variation of (,
steadily decreases, and therefore that each of the magnitudes
remains finite within the range from % "^o — oo . Hence each of
the quantities
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322 liapounoff's [102.
remains finite within tlie range of t from („ to — oo , for a value
of X sufficiently large to make the quadratic form positive.
Lastly, let the constants be complex, so that T, Ti,... can have
complex values ; and now let t be complex in such a way that, in
the variation from to towards — aD , where
( = ff, j re"*,
a remains unaltered. The independent variable now is t, a real
quantity, varying from to — qo ; and the preceding argument
applies, A finite number \ can be found such that each of the
quantities
remains finite within the range of (. But
hence a finite quantity X' can be chosen, so that each of the
quantities
remains finite within the range of ( from („ towards  co .
In the first and the second cases, let
^ = X + (T,
where rr is any real positive quantity that is not iniinitesimal ; and
in the third case, let
where <r is any real positive quantity that is not infinitesimal.
Then, because
in the respective cases tend to zero, as t becomes infinite in its
assigned range, it follows that a quantity //. of finite modulus can
be obtained, such that
all become zero when t becomes infinite in its assigned range.
This is true, a fortiori, when fi. is replaced by another quantity
of the same argument and greater modulus.
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102.] THEOREM 323
lb also is true when any one (or any number) of the quantities
T ghoiild happen to be multipUed by a polynomial in t. For all
that is necessary is to take a value /i+p, where p has the same
argument as (i ; then
e'"P,
where P is a polynomial in t, is zero in the limit, when ( is infinite
in its assigned range. Thus a quantity /i can be chosen so that
where P, Pj, .... Pi_i are polynomials, all become zero when t
becomes infinite in its assigned range from 4, which is not a
singularity of the equation, to — co ,
103. This result is now to be applied to the equation which
determines T. Let t=dr be any one of the roots of Ug ~ 0, and
consider a fundamental system of integrals in that vicinity. If
[ JJ 1
: + (fcl) = U77 1.
the indieial equation for Or is
Suppose that p is not an integer. The integrals which belong to
the exponents 0, 1 , .... k2 are holoraorphic functions of t— 0^
in the vicinity of 0^ (Ex. 12, § 40) ; and the integer which belongs
to p is of the form
(t  e,y F (t  Or),
where P is a holomorphic function of its argument.
The contour of integration has yet to be settled. In con
nection with the value 0^, we draw a straight line from that point
towards — co , either parallel to the axis of real quantities by
preference, or not deviating far from that pai'allel, choosing the
direction so that the line does not pass through, or infinitesimally
near, any of the other roots of [Tj = ; and we draw a circle with
6r as centre, of such a radius that no one of those other roots lies
within or upon the circumference. The path of t is made to be
(i) in the line from — co towards 0^, as far as the circumference of
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324 DISCUSSION OF [103,
the circle, (ii) then the complete circumference of the circle,
described positively, (iii) then in the line from the circumference
back towards — oo . So far as concerns the conditions imposed
upon T by the relation
at the limits, we have only to take the values at the two extremi
ties (=  oo . Now V^ is a linear ftinction oi T,T^, ..., Tt, , the
coefficients of these quantities in that linear function being poly
nomials in z and (; hence, taking z as equal to the quantity ^ of
the preceding investigation, or as equal to any other quantity of
the same argument as fj. and with a greater modulus, we have
at each of the two infinities for ( ; and so the conditions at the
limits are satisfied.
In these circumstances, the complete primitive of the equation
for T is
T^A(t~e;yF{te,) + Qiter),
where Q is a holomorphic function of t — dy, involving m — 1
arbitrary constants linearly. The corresponding integral of the
original equation then arises in the form
Tdt,
taken round the chosen contour.
104. We proceed to discuss this integral for large values
of s. Let a be the radius of the circle in the contour, so that
the series P and Q converge for values of t such that \t — 0r\<a.
For simplicity of statement, we shall assume* that the duplicated
rectilinear pai't of the contour passes parallel to the axis of real
quantities from t^dr — a to i = — x. From the nature of the
integral T, we know that a finite positive quantity X exists, such
that the value of
* The alternative would be merely to take
it value of a, Knii then jnalie t" vary from  a to m .
f""
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104.] THE DEFINITE INTEGRAL 325
remaina finite, as t decreases from 8r — ct to — co . Let £ denote
the maximum value within this range ; then
for all the values of (, and then
Let z have the same argument* as X, and have a modulus greater
than \\\, that is, with the present hypothesis, let z be positive;
then the part corresponding to the lower limit is zero, and we
have
r"'e"rdt < ^ e<'» '"'"• ,
J a, z — x
for values of z that have the same argument as X and have a
modulus greater than X ; aud S is a finite quantity.
Similarly, if, after t has described the circle, S' denote the
maximum value of e^*2' for fl^ — a >(> — <», thou the second
description of the linear part of the contour gives an integral,
such that
r "e'^Tdt < ^^ e''» '^"' ,
for similar values of e ; and B' is a finite quantity.
If, then, these two parts of the integral be denoted by /' and
/"' respectively, we have
where a is a positive quantity ; hence for any constant quantity q,
however large, we have
Limit (s^e'^'I') = 0,
when s tends to an infinitely large positive value. Similarly, in
the same circumstances, we have
Limit (s^e^^'F") = 0.
* This form of atatemeiit is suited also for the variation of ( indicatecl in the
preceding note.
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326 poincare's discussion [104.
Now consider the integral round the circular part of the
contour. As Q(t — 8r) 's a holoraorphic function over the whole
of the circle, we have
taken round the circle ; and therefore the portion of the integral
I e''Tdt contributed by this part is /", where
/" = J (( _ e,)^ e'^P ((  6,) dt,
on taking A — 1. The function P is holomorphic everywhere
within and on the circumference, so that we may take
P(tOr) = c, + c,it$r) + ... + c^(terT + R„,
where \R,a\ can be made as small as we please by Bufficiently
increasing m ; for if g be the radius of convergence of P{t~ 8,),
so that g>a, and if M denote the greatest value of \P{t~8r)\
within or on the circumference of a circle of radius c, where
ii™<i/'
for values of ( such that
it$,\^a<c
The value of the integral taken round the circu inference can
be obtained as follows. Draw an infinitesimal circle with $,. as
centre, and make a section in the plane from the circumference of
this circle to that of the outer circle of radius a along the linear
direction in which t decreases towards — co . The subject of inte
gration is holomorphic over the area of this slit ring : and there
fore the integral taken round the complete boundary is zero. Let
' T. F., % 23.
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104.] OF LAPLACE'S INTEGBAL 327
J' denote the value along the upper side of the slit, J" the value
along the lower side, K the value round the small circle which is
described negatively ; so that
J' = r {i  OrY e"P (*  Br) dt,
J" = /"'■"""esi. (f _ OrY e'=P ((  dr) dt
and, if the real part of p be greater than — 1, then*
Hence, beginning at the point on the outer circumference which
is on the lower edge of the slit, we have
r + J' + K + J" = 0,
that is.
Let u denote the integral
U= { '' it OrY ^^i,
and consider the value of it, for large values of z. Let
(  f , =  T = re" ;
then
Taking real positive values of s, write
so that, as z is to have very large values, the upper limit for %
with the new variable is effectively + oo ; thus
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328 Laplace's solution [104.
Also, if V denote the integral
then
V ^ { ly e^' r T^ e" R^dr
Further,
IJ*^ y'eyR.ndy \ < M (^J"^' ^/„ V^'^^y
which, when the real part of p + 1 is positive, can be made less
than any assigned finite quantity as m increases without limit,
because a < c.
Using these results, we have
r = f' J 2 c„ (( ~ BrY + H J (f  dry e" dt
when TO is made as large as we please, and the real part of p is
greater than — 1. Hence /" is a constant multiple of this
quantity.
105, If now Wr i^enote the integral of the original equation,
we have
■Wr=U'Tdt
= /■ + /" + /'",
so that
For very large values of z, the first term on the righthand side
tends to the value nero ; so also does the second term. The third
is a constant multiple of
i {\yz'c^Vip + a + \).
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105.] AND NORMAL INTEGRALS 323
Hence, dropping the constant factor, we have
Wr = e'^^s'' i ( l)sc. r (p + a + 1),
for very large values of s. If the coefficients, of which c^ T(p+a+l)
is the type, constitute a converging series, then this expression
has a functional significance. If they constitute a diverging
series, the result is illusory from the functional point of view.
Now we have
P + l =
dU,
and therefore the preceding integral, when it exists, is of the
form
When the aeries converges, this expression agrees with the foi'm
in § 100, which is
where <l>r is a holomorphic function of a"' for large valnes of 2.
It thus appears that, when Laplace's solution of the equation,
originally obtained as a definite integral, can be expressed ex
plicitly aa a function of z, which is valid for large values of ^, it
becomes a normal integral of the equation.
This normal integral has arisen through the consideration of
the root S^ of the equation U^ = 0. When the corresponding con
ditions are satisfied for any other root of that equation, there is a
normal integral associated with that root. Hence, when n normal
integrals exist, they can be associated with the roots of the
equation U^ = 0, which comprise ail the finite singularities of the
equation in T.
Note. It has been assumed that p is not an integer. When
p is an integer, logarithms may enter into the expression of the
primitive of the equation for T, and they must enter if p has any
one of the values 0, 1, .... k — 2. There is a corresponding inves
tigation, which leads from the definite integral to the explicit
expression as a normal integral. When the normal integral exists,
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330 EXAMPLES [105.
it can always be obtained by the process in § 100. If logarithms
enter into the expression of e^u, they enter into the expression of
u in the usual mode of constructing the regular integrals of the
equation satisfied by !(.
Es:. 1. The preceding method of obtaining the normal integral gives a
test as to the convergence of the series in its espr^sion. If the infinite
converges, which must be the case if the expression for the developed definite
integral is not to prove illusory, its radius of convergence r is given by the
relation*
Limc„r(p + a + l)r = ^.
But, from Stirling's theorem for the approximation to the value of r In),
when n is infinitely large, we have
%+c^{t6;,+c,,{te;f\...
must conven^e over the whole of the iplane ; and therefore the integral TV is
of the form
wtere ^ (() is holomorphic over the whole plane ; a I'esult due to Poincard
Ex. 2. Prove that, if the condition in Ex. 1 ia satisfied, a normal int^ral
certainly exists. (Poineard.)
Ex. 3. Consider Bessel's equation
for large values of i;. The int^rala in the vicmity of j; = oo may be
normal — they are not regular — and, if normal, must bo of gradi" unity.
Accordingly, let
then the equation for a is
We take
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105.] bessel's equation 3
and theu seek for a regular integral (if any) of the equation
j^m" + {^ + 20^) u' + (to v?)u = 0.
If an integral, regular in the vicinity of a: = =o, can exist, it is of the form
Substituting, and making the coefficients in the resulting equation vanish,
we have
c^{26p + 6) = 0,
anii, for all values of m,
«=j + So„, + i(2p2ml)0.
and the latter then gives
Hence, taking 60 = 1, and 6 = i, a formal solutioa of the original equation is
and taking S= — i, C,=l, another formal solution is
If 2ji is an odd integer, positive or negative, both series terminate ; and the
forma] solutions constitute two normal integrals of the equation. It is not
difficult to obtain an espression given by Lomaiel* for ■/„, in a form that
is the equivalent of
IfSn. is not an odd int^er, both aeries divei^o ; and the formal solutions are
then illusory as functional solutions.
When Laplace's method of solution is adopted, so as to give an integral of
the form
the equation for T is
On writing
(i^ + l) T" + ZtT' + {l ^ n^) T=0.
V independent variable, the equation for T is
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BB8SELS EQUATION AND
[105.
This is the differential equatioi
elements are given by
e function, wliose (Gau
o + fi + l=3
a^=l
Thp contoui cf the integral oonsi^tR of (i) a <,irde ruund i as tPiitre with
radius leas tlnti 2 ('o as to exclude — ;, the otlier finite singuHnty of the
equation m T), and then (ii) a duphcited hue fium a point in the Liruum
feience pissing m the direction of a diameter tontinued towards co The
ai^ument of / and the argument of j: muit he ^uth that the real pajt of xt
is negative. In order to construct the integral, we need thf complete primi
tive of the ^equation in the vicinity of ii = 0: it is
where A and B are arbitrary constants. The part multiplying A, being a
holomorphic function, merely contributes a zero term to w ; and we need
therefore substitute only the other part. Manifestly, we may write B=l,
Now
^(ay + 1, j3y+l, 2y, ^) = i^(<ii, (3^, h ")
1 2m + l
, „ 2m +
n(m+^)n(^);
Taking this value of c^,, we substitute
in the definite integral. In the preceding notation, we have
d.=i, p=i, <r,= (p + l)=i
n(m<r,) = n(m + 4);
so that, when the solution
where t=i2iv, is expanded into explicit form, ifc becomes a constant
multiple of
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105.] DEFINITE INTEGRALS 333
But
^n(,H)^ "«'"'»':l'"*'''' n(i).
SO that, after substituting for c« and rejecting the constant factor n(^),
the integral becomes a constant multiple of
m=o ni! {2iJ^)""
which agrees formally with the expression earlier obtained.
The corresponding integral, associated with the primitive of the ^equation
in the vicinity of i= — i as a singularity, can be similarly deduced*.
E^: 4. Shew that the equation
where «,'^  ia^ is not zero, can be transformed to
xy/' + {\, + \ + 2)'y/ + {x+i (\i^\ti)}w = 0.
Assuming Xi, X^i ^i + ^a "^ot to be integers, prove that the latter equation is
satisfied by
for an appropriate contour independent of x ; and deduce the normal series
which formally satisfy the equation. (Horn.)
Doubleloop Integrals.
106. Before proceeding further with the investigation in
§§ 101 — 105, which is concerned partly with the precise determ
ination of a definite integral satisfying the linear differential
equation, we shall interrupt the argument, in order to mention
another application of deiinite integrals to the solution of certain
classes of linear equations. It is due to Jordanf and to Poch
hammerj, who appear to have devised it independently of one
' In connection with the aolation of Bessel's equsition by mes^ne of definite
integtals, papers by Hunkel, Math. Ann., t. i (1869], pp. i67 — 501; Weber, ib.,
t. 3X3V1I (1890), pp. 404—416; Macdonald, Proc. Lond. Math. Soc, t. sm (1898),
pp. 110—116, ib., t. XXI (1899), pp. 165—179; and the treatise by Graf u. Gubler,
Einleitung in die Theorie der Bessehchen FunUionen, (Bern), t. i (1898), t. n (1900) ;
may be consnlted.
t CoiiTB d'Aiwlyu, 2= k&., t. Ill (1896), pp. 240—276; it had appeared in the
earlier edition of this work.
+ Math. Ann., t. xsxv (1890), pp. 470—494, 495 536; II., t. sxxvii (18S0),
pp. 500—511.
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334 nouBLELoop [106.
another. A brief sketch is all that will be given here : fur details
and for applications, reference may be made to the sources just
quoted, and to a memoir by Hobson*, who gives an extensive
application of the method to harmonic analysis.
As indicated by Jordan, the method is most directly useful in
connection with an equation of the form
where Q (s) and zR (s) are polynomials, one of degree n, the other
of degree ^niiiz, R {z) also being a polynomial. For simplicity,
we shall assume Q (s) to be of degree n.
Consider an integral
W= { T {t  zY+''' dt,
where 2" is a function of t alone ; this function of ( has to be
determined, as well as the path of integration. We have
AW(l)"(a + 7il)(a + n 2). ..(« + ])
+ {tzY\R{z) + {tz)R'{z)^^^''R"{z)+JATdt
= /[« (*  ^y' Q (i) +(tzrii im Tdt,
the summation being possible because Q and R are polynomials
of the specified degrees. The integral will be capable of simplifi
cation, if the integrand is a perfect differential ; accordingly,
we choose T so that
which gives
TE(t)=^[TQ{t)\,
Q(t)
* Phil. Tmne., 1896 (A), pp. 443—531.
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106.] INTEGRALS 335
The preceding integral then becomes
jdV,
where
mt)
Hence the original differential equation will be satisfied if
jdV^O;
and this will be the case, if the path of integration is either
(i) a closed contour such that the initial and the final values
of V are the same : or
(ii) a line, not a closed contour, such that V vanishes at each
extremity*.
Each such distinct path of integration gives an integral. It ia
proved by Jordan that there is a path of the first kind, for each
root of Q ; and that, when there is a multiple root of Q, paths of
the second kind are to be used.
Again, restricting Q (s) for the sake of simplicity, we assume
that each of its n zeros is simple; let them be (ti, esg, ..., a^ As
the polynomial R (z) is of degree less than n, we have
M(fl
= S 
where 7,, ..., 7,1 are constants; and then
To obtain the paths desired, take any initial point in the plane ;
from it, draw loops^f round the points a,, ..., On, z, and denote
these by A^, A^, ..., An, Z. Take any determination of
* A third possibility would arise, if the patli were
same value at its extremities; tut thia case is of yery n
■t T. F.. % 90.
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DOUBLELOOP INTEGRALS [106.
which is the subject of integration in W, as an initial value ; and
let the values of W, for the various loops J.,, ..., j4„, Z with this
as the initial value, be denoted by W {a^, ..., W{a^, W{z)
respectively.
An integral of the original differentia] equation will be ob
tained, if the path of integration gives to F a final value the
same as its initial value. Such a path can be made up of
ArAgAr~'^Ag^, that is, first the loop Af, then the loop Ait, then
the loop Ar reversed, then the loop jig reversed. Let W{ar, Wj)
denote the value of the integral for this path ; then W{aT, a^ is
a solution of the differentia! equation. Taking the above initial
vahie (say /(,) for /, we have
W{ar, a,) = W(ar) + e"^>r Wia,)  e="^*. W(ar)  W(a,)
= {1~ e^'v,} W(ar)  {1  e^r] W (a,) ;
for after the description of A^, the initial value of 7 is e^'^t./^
for the description of Ag, it is e''"'>'r+''s'7t, for the description of
Ar~', and it is e""^*./,, for the description of j1,~\
It is clear that
[le'^yi] W (a„ ar) ^ [1  e"^'.} W (a„ at) + {1 ~ e'"^,] W{at,ar);
and therefore al! these values of the integrals, for the various
appropriate paths, can be expressed linearly in terms of any n of
the quantities W(«r, Cs)> in particular, in terms of
W(z,a,). W(z,a,), ..., W(z,a^).
Each such quantity is an integral of the original equation ; and
we therefore have n integrals of that equation.
iVofe. For the special cases when a or any of the conetantB y is an
int^er ; for the cases when § (I) has multiple roots ; and for the oases when
B(t) ia of degree n — \, while Q{1) is of degree less than m — 1 ; reference may
be made to the authorities previously cited. As already stated, all that is
given here is merely a brief indication of the method of doubleloop integrals.
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106.] EXAMPLES 337
Es:. 1. Consider the equation of the qu.irterpei'iod in elliptic functions,
Here we have
9W«(l),
/'
t''{trf=r'{t\)'^
■(iiy^(tz)'dl,
and tho path of integration has to be settled.
We have
W(lj==2J'dW,
I dW,
where a mai'ks the initial point of the loops. Hence
H'(0, l) = 2W{0)2W{l) = ij' dW,
W[0, 02tf(0)2F{2) = 4prfir;
and thu.s two integrals of the equation are given by
l^dW, tdW.
The comparison with the known results is immediate.
Ea;. 2. Integrate in tho same way the equation
,, „, rf% ^ dw ,
' ' cb= dz
whore a and 6 are constants. (This is another form of the equation
(l,.)2'2(»+l).f+(.«)(»+« + l),.0,
by Hobson {I.e.) for unrestricted values of the constants m and re.)
V. 22
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338 ASYMPTOTIC! [106.
Ex. ;i. Prove that when the equation
wheie /( i» I tuuittnt, 15 iiiiijectpd to thf tr wistorm xtion
the tunstonned equation (nhich is of Fuchsnn ty]ie, § 54) can, under a
LCitain condition, be trcat<yl bj the foregoing method ind issuming the
condition to be ^atiified ohtaiii the mtegial
Ej 4 ^pply the method to the equation
apply it alao to the equation of the hypergeometric series.
(Jordan ; Pochhammor.)
Ex. 5, Apply the method to solve the equation
{\i')ie"2zw'lw = 0,
for real values of s such that — 1 <3<1. Sliew that the equation is tiaus
formed into itself by tiie relations
(^l)(irI) = 4, w{z + \f^W{Z+lf
and deduce the solution for real values of z such that 1 < z < gd .
(Math. Trip., Part ir, 1900.)
PoiNCABit's Asymptotic Representations of an Integral.
107. After this digression, we resume the consideration of
the investigations in §§ 101—105.
In those ca.ies wheu the infinite series in a normal integral
diverges, the normal integral has been rejected as illusory from
the functional point of view. There are, however, cases belonging
to a general class which, vfhile certainly illusory as functions of
the variable, are still of considerable use in another aspect : they
are asymptotic representations of the integral, to use Poincar^'s
phrase*.
A diverging series of the form
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107.] KEPRESENTATIONS 339
is said to represent a function J(x) asymptotically when, if 8n
denote the snm of tho first ra + 1 terms, the quantity
„•{!(,) S.\
tends towards zero when x increases indefinitely ; so that, when a;
is sufficiently large, we have fl;"{i/"(a:) — S„) < e, where e is a
small quantity. The error, committed in taking S„ as the value
of J, is less than
Lch smaller than
that is, the error in taking Sn as the value is much smaller than
in taking S„_i. (The definition, though stated only for large
values of x, applies also to the vicinity of any point in the finite
part of the plane, tnatatis mutandis.)
The asymptotic representation is, however, not effective
for all values of the argument of the independent vaiiable. If
a;" {/(«)  Sn) tended uniformly to zero for all infinitely large
values of x. the function J(x) would be holomorphic, and the
series would converge: the permissible values of the argument
of the independent variable are therefore restricted. It is manifest
from the nature of the ease that, when such a series is an
asymptotic representation of a function, the series can be used
for the numerical calculation of the approximate value of the
function for large values of a: with a permissible argument: the
error at any stage is much less than the magnitude of the term
last included. Without entering upon any discussion of the
question why a diverging series, which is functionally invalid,
can yet, when it is an asymptotic representation of a function,
be of utility for the numerical calculation of the function, it is
proper to mention one conspicuous example of the use of such
series, as found in their application to dynamical astronomy*.
The normal series, derived from the solution of the equation
as represented accurately by the definite integrals, are proved by
Poincar^ to give this type of asymptotic representation of the
* In particular, see Poinoare, HUcaniqiie Cileste, t. ii.
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340 NORMAL INTEGRALS AND [107.
solution. For, denoting the solution by w, and tlie sum of the
first m t 1 terms of the series
by Siji, we have
Now
c
where
and
iHl<l.
Then, as before, we have
r M u
7 0™+' J _liPr
c
which is a multiple of
by a quantity independent of a. When we take
so that, as a is to have large values, the limits of y effectively ai'e
to + CO , the last definite integral is a multiple of
I T — •T^y'*''^"^eydy.
This definite integral is finite. Denoting its value by 7, we have
where a is a quantity independent of e, and 7 is finite. Hence,
when 2 is sufficiently large, we have
z™ (we~=*'s''+'  Sm) < e,
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107.] ASYMPTOTIC EXPANSIONS 341
where  e  is a small quiintity ; and so we can say that S^ asym
ptotically represents we~'^' £'•+'■, or we can say that the normal series
is an asymptotic representation of the actual integral, the repre
sentation being valid (on the hypotheses adopted earlier) foi' large
positive real values of ^.
Note. For further diseuasioii of these asymptotic expansions
in connection with linear differential equations, reference may be
made to Poincar^'a memoir*, which initiated the idea. Among
other memoirs, in which the subject is developed and new applica
tion.9 are maile, special mention should be made of those^f by
Kneser, and thosell by Horn. Picard's chapterj on the subject
may also be consulted with advantage: and a corresponding dis
cussion on integration by defiuite integrals is given by Jordan§.
Ex, 1. Shew that the complete primitive of the differential equation
in the vicinity of ;c = co , can be asymptotically represented by
('^ + "' + ,^^+)«>«'^+(^o+t + 5 + )^'"'*''
and hq, (9o ive arbitrary constants. (Kneaer.)
Ex. 2. In the differential equation
i(*S)+(«+<')»°
ifc^ is an arbitrary parameter, A, B, C are real functions of x and (with their
derivatives) are holomorphic when ai^x^h; moreover, A and B are positive.
Prove that a,ii int^ral of the equation, determined by initial values that are
independent of h, is a holomorphic traoseeiidental function of k ; and shew
that, for large values of k, its asymptotic expansion if of the form
,!,.(*,+*>+.^.)co.i,+(*l + J' + .,.).mi»,
whore ^a, (f",, <^j, .,., mi are functions of x, (Horn.)
* Acta Math., t. vni (1886, pp. 295—344,
+ Crelle, t. oxvi (1896), pp. 178—213; ib., t. txvii (1897), pp 72— lOS; lb.,
t CM (1899), pp. 267—275 ; Math. Ann., t. xlix (1897), pp. 383—399.
II Math. Ann., t. xm (1897), pp. 432—473, 473—496 ; i6., t. l (1898). pp. 525—
556, ib., t. LI (1899), pp. 346—368; ib., t. LH (1899), pp. 371—392, 340362.
X Coura d'Analyse, i. iii, ch. xiv.
% Goiirs d'Analyse, t. iii, ch, n, § iv.
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342 RANK [107.
Ex. 3. Shew thiit the equation
has a solution of tlie form.
where A^, Bj„ ace rational functions of k, and that it has an aaymptotic
solution of the form
(*.+= + ...)o™fa+(^ + *J + ...).mfa,
and indicate the relation of the Molutions to uue anofclier, (Poinearri : Horn.)
Equations of Rank greater than Unity replaced by
Equations of Rank Unity.
108. When the differential equation
possesses, in the vicinity of 3 = O) , normal integrals which are of
grade m, then, denoting the degree of the polynomial p^ by ■m^, it
follows (aa in § 85) that the degree m,. of the polynomial p, is such
that
the sign of equality holding for some at least of the <
Also, if e" be the determining factor of any such integral, then
Of is the aggregate of the first m terms in the expansion, in
descending powers of z, of a root of the equation
The existence of the normal integral then depends upon the
possesion of regular integrals by the linear equation in u, where
In the case where in = 1, the method of Laplace certainly gives
the integrals of the differential equation, even wheu the normal
series diverge ; but it is not applicable, when m is greater than
unity. Poincare, however, devised a method by which the given
equation is associated with an equation of grade unity: Laplace's
method is applicable to the new equation, so that its primitive is
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108.] OF EQUATIONS 343
known : and from tliis primitive, an integral of the original
equation can be obtained by means of one quadrature. The new
equation is of order ra'"; and the investigation leads to an
expression for
w dz '
which, whefl it oxiata, can be obtained more directly by Cayley's
process (§ 92),
Poinear^'s method is as follows. Let the given equation be
supposed to possess n normal integrals of grade m, say, in the
form
e"'<''.^(4 B"'^'>M^) e""'«0«(^);
let these be denoted by /i (a), /^(j), ...,/„ (a).
Let a denote a primitive mth root of unity, say e™ ; and
consider, in connection with any integral /(z) of the original
equation, a product
!,="nV(«'^).
Then y satisfies an equation of order ii™. which possesses ?^™
normal integrals
/.(')/. («)/.(»'^) .■•/.(«H
where a, b, c, ..., k are the numbers 1, 2, ..., n or some of them,
any number of repetitions being permitted ; and these normal
integrals are of grade m. Lot
and let the equation for y be
where, if Q^ be of degree in z, then the degree of Qjc_r in
general is equal to d + r{m—'l), because of the grade of the
normal integrals. Owing to the source of the quantity y, which
clearly is not changed if 2 be replaced by sa*, s being any integer,
it follows that the equation for y must remain substantially
unchanged, when this change of variable is made ; hence
fcr(s«')g''^'"" _
where \ is independent of r.
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344 BANK CH. .>'GED [108.
Now let the variable be changed from z to x, where
then, because
for all values ef «, the coefficients c„. being numerical, the equation
for 2/ takes the form
■where
The degree of fi^yj in ^, as it is determined by the highest terms
in Q,va. i*5
which is independent of 3; so that the degree* of all the
coefficients R is the same. Further, wc have
B,_, (.«■)  i^ c„„ («')'>'.''>'e„_(««')
for the power of a is
thus
Hence the equation is substantially unaltered, when z is replaced
by 20* in the coefficients Jt ; hence, multiplying by a power of s,
say z", where
« 4 f + if (m  1) = (mod m),
i£ becomes a uniform function of x, when we substitute
" Some mijht haye vanishing ooelflcieiits in particular cases; the argument
deals with the genei'al ease.
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108,] HY TRANSFORMATION 345
The new equation is therefore an equation in the independent
variable x such that all its coefficients are uniform, They ail are
of the same degree, so that it is of rank unity ; it has normal
integrals, and some of its integrals may he subnormal. Laplace's
method can be applied to this equation ; and we then have a
solution in the form of a definite integral.
The way in which this definite integral is used, in order to
bring us nearer a solution of the original equation, is as follows.
Let
'.=/(^a").
(s = 0, 1, .
1),
This has to be differentiated N(=n™) times, derivatives of w„,
Wj, ..., w,„_i of order n being replaced, whenever they occur, by
their values in terms of derivatives of lower order, as given by
the diffei'ential equations which they satisfy; and, from the iV"+ 1
equations involving y, j^ , ,.., j^, the iV" products
de^ '"rfs* ■
d^
where a, b, ..., k «
1 can have the values 0, 1, .
,. , ., ..., k each can have the values 0,
eliminated. The result is the equation for y.
involving y, ^, ..., ^E^ can be regarded
products of the type
The N equations
i giving these N
"^~d^
each in I
such be
i of derivatives of y and the variables. Let two
Assuming p known, as an integral of its own equation, the value
of lUo is derivable by a quadrature. If y, first obtained as a
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346 POINCAR^'S METHOD [108.
definite integral, can be evaluated into a functionally valid normal
integral, it is of the form
The function $ is linear in y and the derivatives of y, so that,
when we substitute the value of y, we have
ivhere "I* is^ free from exponentials ; and then
1 dw^ _^
which can be expressed as a series in terms of z. The exponent
to which it belongs is easily seen to be an integer, owing to the
form of ■!> ; thus
1 <^M;n ™ , ^, „ a,„ «„,+i
j=(l(,3"'' + ((l2™^+ ... +(tmi+ — H >+■■■
Wa dz Z Z^
But if y cannot be evaluated into a functionally valid normal
integral, there may be insuperable difficulty in dealing with the
quantity — .
In instances, where the actual expression of a normal integral
{if it exists) is desired, the process is manifestly cumbrous: as
it does not lead to explicit tests for the existence of normal
integrals, the simpler plan is to adopt the process indicated in
§ 85 — 88, which gives either a normal integral or an asymptotic
expression for an integral in the form of a normal series.
For further consideration of Poincare's method, reference may
be made to his memoir, already quoted, and to a memoir by
Horn*, who discusses in some detail the case, when the linear
equation is of the second order and of rank p.
Ex. 1. In the case of an equation of the second oider which is of rank 2,
saj
shew that, if w = 0(«), and if w^i^^{^x), which will satisfy the equation
d^w, , , , dw,
i«^><'>7Er+''><"''"'»'
 Asia Math., t. xxiii (1900), pp. 171—201.
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lOS,] EXAMPLES
then ;i vai'iablo y., whore
uniquely in terms <iiy.
Tf, liowever, the invariants of the two equations are equal, so that
of a quadratic equation, the coefficients of which are expre^ible in terras
of y. (Horn.)
TJx. 2, Discuss the equation
for lai^e values of a. (Poincare.)
Ex. 3. Shew that, in the vicinity of r— o^ , the equation
ormal integral of the second grade, when a is an odd positive
Es^. 4. Ohtain the normal integrals of the equatioi^s
(i) :k^" = (^ + )2/,
(ii) ^y = 2j;(l + &^)y + (;^^6V26a;j:)y,
in the vicinity of a^=o; ,
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CHAPTER VIII.
Infinite Determinants, and their Application to the
Solution of Linear Equations.
109. In the investigations of the present chapter, infinite
determinants occur. These are not discussed, as a I'ule. in books
on determinants ; a brief exposition of their properties will there
fore be given here, but only to the extent required for the
purposes of this chapter. Their first occurrence in connection
with linear differentia! equations is in a memoir* by G. W. Hill :
the convergence of Hill's determinant was first established f by
Poincare. Later, von Koch shewedj that the characteristic method
in Hill's work is applicable to linear differential equations generally;
with this aim, he expounded the principal properties of infinite
detenninants§. The following account is based upon von Koch's
memoirs just quoted, and upon a memoir] by Caazaniga,
Let a douhlyinfioite aggregate of quantities be denoted by
where i, k acquire all integer values between — co and + x ; the
quantities may be real or complex, and they may be uniform
functions of a real or a complex variable. They are set in an
* First published ill 1977; republished 4c (a Math., t. viii (1886), pp. 1—36,
t BvU. de la Sw. Math, de France, t. xiv (1886). pp. 77—90.
I Acta Stath.. t. sv (1891), pp. 53—63 ; ib., t. nvl (1892—3), pp. 217—235.
§ Foi tocther cliscuBBion of theii properties and their applications to linear
differential eiiuations see a memoii by tbe same writer, Aeta Mat},., t. xstv (1901),
pp 89—122
II Anwih di Uai mat a *5er 2' t jcxvi (1897), pp. 143218. Other memoirs
by CazzaniKa lealin^ with the la.iai' subject, are to be found in that journal,
&ei ' t i18JS pp si14 S^ > t. II (1899), pp. 329238.
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109.] INFINITE DETERMINANTS 349
array, so that all the quantities with their first suffix the same
occur in a line, the values of k increasing from left to right, and
all the quantities with their second suffix the same occur in a
column, the values of i increasing from top to hottoni. We then
have an infinite determinant, which may be represented in the
form
Constrnct the determinant An.n. where
then if, as m and n increase indefinitely and without limit, i),n,»
tends to a unique definite value D, we regard the infinite determ
inant as converging to the value D. In all other cases, the
infinite determinant diverges. To secure this convergence to a
unique definite value D, it is sufficient that, when any arbitrary
small quantity S has been assigned, positive integers M and N can
be found, such that
!0«.+p,n+s0™,«<s,
for all values of m greater than M. for all values of n greater than
iV^, and for all positive integers p and q.
The aggregate of all the quantities for which i = k, that is, of the
quantities ..,, a_i,_i, (X„,o, Ch.i, as they occur in their place in the
determinant, is called the principal diagonal, sometimes briefly
the diagonal ; and a constituent of reference in the diagonal,
naturally chosen in the first instance to be aj.u. is called the
origiv.
Let
then the infinite determinant converges, if the doubly infinite
converges, all values of i and k between — <» and + oo occurring ii
the summation. To prove this, let
p„..= n 1+ s 4,.,i
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350 COUVEllGENCE OF [109.
and consider
Let P,„,B be expanded ; by omitting suitable terms and chauging
the signs of others, we obtain Dm,». Hence, taking I'm,.!, making
al! the terms positive, and adding certain other positive terms,
we obtain .?„,„. Similarly^ we can pass from D^^p^n+q to
Pm+p,n+q Now take i>m+p,n+9 — Om,!! ; make all the terms
positive, and add certain other positive terms, and we have
i).+,.„^,A.,«<P™..,«H,^..,«l
But, because of the convergence of the series
the product P^.n converges when m and n increase without limit ;
hence, assuming any arbitrary positive quantity S, however small,
integers M and iV" can be determined such that
Pmtp,n+q  P'm,n < ^i
for all values of to greater than M, for all values of n greater than
N, and for all positive integers p and q. Consequently, for the
same integers, we have
and therefore the infinite determinant converges.
Such a determinant is said* to be of the normal form. AU
the determinants with which we have to deal are of this type.
Next, the origin may be changed in the diagonal without
affecting the value of the determinant. All the conditions for
the convergence of the determinant with the new origin are
satisfied; let its value be D', and let D be the value with the
old origin. Then taking any small positive quantity 5, we can
determine integers M and N such that
t'Om,«<S, i>'i>V,„,<^,
' von Koch, Acta Math., t. xvr, p, 221.
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109.] INFINITE DETERMINANTS 351
for all values of m greater than M and all values of n greater than
N, the determinant D'^^n, being the same as il^^n, ao that, if agg
be the new origin, m, = m $, Wi = n + ft Manifestly, D^^^ can
be chosen so as to include the new origin. Hence
\Dn'\ = jD  »™,„ (D'  Z>V,,„)
<\DD,„„\ + 'D'})\„.,\
so that, in the limit when 8 is made infinitesimal,
d = b:
Similarly, the value of the determinant changos its sign when
two lines are interchanged, and also when two columns are inter
changed: so that, if two lines be the same, or if two columns be
the same, the determinant vanishes. Further, if the determinant
be changed, so that the lines (in their proper order) become
columns and the columns (in their proper order) become lines,
the principal diagonal being unchanged, the value of the determ
inant remains unaltered. If, in any line in a determinant of
normal form, each of the constituents be multiplied by any
quantity ft, the value of the determinant is multiplied by /j. ;
likewise for any column, and for any number of lines and
columns, provided that the product of all the factors (when
unlimited in number) converges.
Further, if all the constituents in any line of a converging
normal determinant be replaced by a set of quantities of modulus
not greater than any assigned finite quantity, the new determinant
converges. In the determinant D, let the line ao^j (the constitu
ents occurring for values of k) be changed, so that a^^j is replaced
by *'t, where
kll < A,
A being finite ; and let D', !>'„_ a for the new determinant
correspond to D, D„^„. For comparison with Z*'„,„ construct
a product P,„,,(, where
p,.,,..'n'i + l^,.,i,
1 having all values from —n to +m, except i = 0. Then, when
i>',„_,i is expanded, there occurs in P^.n Jt term corresponding to
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352 PKOPBKTrES OF CONVERGING [109.
every term in D'm^n. tbe latter having gome one factor x^ that
does not occur in ^m.n\ hence
I term in i?;„,„^^ jterm in P™,„.
Now some of the terms in D'™,„ are negative, while all the terms
in Pm,n. are positive; and terms arise in Pm,n. the terms corre
sponding to which do not occur in D'm,n Hence
Similarly,
where h can he chosen as small as we please, becanse
i i + I \AiA
is a converging proiluct.
The resnlt, which is due to Poincare, is thus established.
Properties of Oonvergisg Infinite Determinants.
110. The development of an infinite determinant can be
deduced from the preceding properties. We have
n I "tji, 11+1 • ■■■! 'hn.t
= 2 ± (»_„__.„«._„ ^.,,_„+l . . . am.ni)
say. In this expanded form, let
ai_i = l + Aij, ai^k'^^w (' + *);
and let every term in the new expression be changed, so as to
have a positive sign and so that each factor is replaced by its
modulus. The resulting expression is greater than Sm,„; and
every term that occurs in it is contained in P,„,«, where
_ m ( m )
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110.] INFINITE DETEEMINANTS 353
Also, Pm,« contains other terms, all of which are positive ; thus
S„,,<P,..,.
Similarly, _ _
for all positive integers p and q. But P,„,„, with indefinite
increase of m and m, is a converging product; hence 2m,n. in
the same limiting circumstances, converges absolutely. Thus the
usual method of development of a finite determinant holds in the
case of an infinite converging determinant of the normal form, and
«[»»] l]Z}
= 2...a_2,j,,tt_,,p,ao,y,aj,g,as,g^ ...
(_lY..+ (Pi2) + (J'i" + (J>o"> + Wiil + i5)"l+
the sura being extended over all the permutations
• ■■. p.., Pi, p«, ?i. 9^, ■■■
of the integers
.... 2, 1, 0, 1, 2, .,..
Writing
for all values of i and k, we at once have the expansion
Z) = l + 2Ji,( + 2^(,i, Aij\ + t\Ai_i, Ai,j, ^i,i ] + ...,
^Aj^i, Aj^jl 4j,i , Jjj, A}^ie\
the summations being for all integer values from — gc to + co such
that
i<j<k<....
111. It follows from the preceding expansion of a converging
determinant D of normal form that, when a constituent o^j enters
into any term of the expanded form, no other constituent from
the line i or from the column k enters into that term. Taking
the aggregate of terms {each with its proper sign) into which Oi^ic
enters, theii sum may be denoted by ra; ^a^j ; and the determinant
may be represented in the form
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354 MINORS [TU.
or in the form
The quantity Wj^i is called the minor of an, and sometimes it i
denoted by
It can be derived from D by suppressing the line i and the
column k, or, what is the equivalent in value, by replacing Oj^j
by 1, and every other constituent in the line i or in the column k
or in both by 0, and then multiplying by (— 1)'"*. Manifestly, we
have
It is an immediate corollary that
0= S aj^tai.k, I
(»4=i)[:
k='B I
for the righthand side in the iirst is equivalent to i> with the
line * replaced by the line j, so that the latter is duplicated ; and
in the second, the righthand side is equivalent to D with the
linej' replaced by the line i, so that the latter is duplicated.
More generally, if, in the lines
and in the columns
ft. A ft,
we replace all the terms by 0, except «a„B,, «o„p,, ., aa,.,e,., each
of which we replace by 1, and then multiply by
the result is the coefficient of
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111.] OF FINITE ORDKB 355
in jD. It nianifcstly is a minor of order r; and it is denoted by
Clearly all the minors of any finite order are determinants of
normal form, converging absolutely.
If D is not zero, some at least of tlie minors of constituents in
any line must be different from zero, and some of the minors of
constituents in any column also must be different from aero.
Similar results, when D ia not zero, hold for the minors of any
order r of finite determinants, which are constructed out of r
selected lines and any r columns, oi' out of r selected columns
and any r lines.
Further, the minor
r + l. ..., 0, 1,
r+1 0, 1,
tends to the value unity, as r and s increase. To prove this, let
Q,,~n{l + X\A„]].
where the product is for all the values of p, and the summation
is for all the values of q, that are excluded from the ranges
p = — r to + s, q — ~rto^s.
Expanding the minor, and changing every term so that its sign is
positive and each fsictor in the term is replaced by its modulus,
we have a new expression every term of which is contained in the
expanded form of Qa,>', and Qg^,. contains other terms. Further,
the expanded minor contains the term +1 as does Q,^,., and all
other terms involve the quantities A ; hence
(::; :::5::;;; 3 i< «..'.
But the product
uh + iiA^A
converges ; and therefore, when any small positive quantity S is
1, integers — r and s can be determined such that
Qs.r  1 < S.
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356 EXl'AJiSION Of [111.
Taking these as the integers defining the minor, we have
80 that
is<(:::::;:;::::;:):<i+s
Moreover, as integers s', r are chosen, greater than s and r and
gradually increasing, the quantity
decreases ; and thns the minor tends to the value unity as r and s
increase.
One or two properties of minors may be noted. We have
\k, l) \k, l) \l, k) \l,k)'
for the changes from one of these expressions to another are
equivalent to an interchange of two lines or an interchange of
two columns, each of which changes the sign of the determinant.
Similarly for minors of any order.
Again, expanding ai^^ by reference to constituents of a column,
we have
and expanding it by reference to constituents of a line, we have
Similarly,
!, because it is
me; also
when q is neither k nor I, because it is a minor of the first order
with two columns the same ; also
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111.] INFINITE DETERMINANTS
whon h is neither * nor j, because it is a i
with two Hnes the same ; and
357
.■ of the first order
\k, I)
= 0.
where h is neither i nor j, and q is neither k nor I, because it is a
minor of the first order with two columns the same and two hnes
the same. Similarly for minors of higher order.
The similarity in properties between finite determinants and
converging infinite determinants of normal form is not exhausted
by the preceding set : in particular, infinite determinants can be
multiplied, and determinants framed from minors of an infinite
determinant are connected with their complementary in the
original, exactly as for finite determinants. The simpler of these
properties are contained in the following examples.
Ex. I. If
are oonvei^ing doterminants of normal type, itnd if
for all values of i and t, then
c[»...l
s a convoi'ging determinant of normal type, and
AB = C.
Ex. 2. If ai,i, denote the minor of o^t in the determinant
%k.' ■■■' %k.
"ik,' *■■' %i; I
with the preceding notation for miDora of order ;■.
Ex. 3. In connection with the determinant
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S5S INFENITE DETERMINANTS AS [111.
prove that
Qfe)+©(t»)^©(*;S(U;l)''^
QO;:S+©(U)*(')(l;i)C:*;:t)"'
and, more generally, that
L f%\fH J H 1 .... ^ \_(hii h, h, ■■■! ir\ A
.iAU\h.i, K*ii, , fni/ W. *i ^, ■. V
where, in the typical term, k^, it„ + i, f« + 2, ..., ifcni preserve the aame cyclical
order as fc^, ifcj, ij, ..., i^
In the first of these, the righthand side vanishes if i is equal to ^, or l;^ i
in the second, it vanishes if i is equal to t, or 4 ; in the third, it vanishes if
k„ is equal to any one of the quantities tj, ^3, ..., i,.; and so in other cases.
112. The infinite determinants which arise in the discussion
of linear differential equations have, as their constituents, functions
of a parameter p. The preceding results are still valid, if the
condition that
is an absolutely converging series is satislned; in particular, the
determinant converges absolutely, and its value may be denoted
by D (p). The parameter may be made to vary ; and then it is
important that the convergence of D{p) should be not merely
absolute, but also uniform, in order that it may be differentiated.
Suppose that, in any region in the pplane, all the functions
Aij(p) are regulai' functions of p, such that the series
converges uniformly and absolutely. For all values of p within
that region, any small quantity B can be assigned, and then
integers M and N exist, such that for all integers m^M, and
integers —n^ — N,
11 '1 A.cj{p)\<&.
By analysis that follows the earlier analysis practically step by
step, wo then infer that, for all integers m'^M, n'^N, and for all
positive integers p and q, and for all values of p within the region
indicated, we have
D,„,,.+,<p)D„,.(p)<28;
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112.] FUNCTIONS OF A PARAMETER 359
SO that l>(p) converges uniformly. Hence, within the domain
considered, I) (p) is a regular analytic function of p.
The expansions of D (p) in terms of its constituents have been
proved to converge absolutely, by comparison with the expansions
of Pm,w. where
converges uniformly and absolutely, Pm.n is a product that con
verges uniformly with indefinite increase of m and n. The
corresponding modifications in the investigation lead to the
conclusion, that the expanded I'orra of TJ(p) converges uniformly
as well as absolutely.
Moreover*, this expanded form can be differentiated, and its
derivatives are the derivatives of D (p). In particular, we have
dp 9ffl,,i dp
= Z2, a. . ^^ .
■ dp
Thus if D vanish for a value p' of p, and if all the first minors of
D vanish for that value, we have
^1 = 0,
while ;r^ is not iniinite; the first derivative of the uniform
dp
function D vanishes, and therefore p' is at least a double zero of
B. In that case, we have
d^D ^ d^ai I .p.^ 9a; 1 3"; t dai ,
jiSo...
dp dp
s^ss(';{
9af,t dojj
Hence, if all the second minors of D vanish for that valui
we have
dp
' The proof is aimilur to those given for preceding propositions ; see to.
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360 INFINITE SYSTEMS [112.
and so p' is at least a triple zero of D. And generally, if all
minors of all orders up to r — 1 inclusive vanish, but not all
minors of order r, when p — p, then p is a root of D in multiplicity
r ; and B is then said to be of characteristic r. The quantity r
cannot increase indefinitely, for we have seen that minors of
sufficiently high order tend to the value unity, so that the general
vanishing of all minors of the same order is possible only for finite
orders.
But it need hardly be pointed out that the converses of these
results are not necessarily true: thus p = p' might be a double
root of D, while not all the first minors of D would vanish.
113. The purpose, for which infinite determinants are to be
used in this place, is in connection with the solution of an un
limited number of equations, linear in an unlimited number of
constants. Let
and suppose that the infinite determinant B, where
converges uniformly ; it is required to find the ratios of the
quantities x to one auother which satisfy the equations
Ui = 0; (i =  ^ to + 00 ),
the quantities a; being themselves finite, so that we have
where X is finite.
We know that
fG)
converges absolutely; its value is B when j = k, and is when /
is different from k. Moreover, the series S<\s is an absolutely
converging series, and hence for values of o! considered, we have
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113.] OF EQUATIONS 361
where TJ is finite. Hence, by one of the propositions already
establishei^, the quantity S, defined by the equation
also converges absolutely, so that
for all the other terms give a zero coefficient for x. Hence, if
Mi = for all values of i, and if we are to have values of ic^ different
from zero, then
D = Q,
which is a necessary condition. We shall assume this condition
to be satisfied.
If some at least of the first minors are different from zero,
then the equation
shews that any one of the quantities u, which it contains, is then
linearly expressible in terms of the others, and so the correspond
ing equation w = is not an independent equation. Let m, then
be omitted on this ground ; we have
?(i,'iV=ffG; ")"'■'''■'■
where on each aide the summation is for all values of i except
1 = 0. The coeificient of x^ on the righthand side is
?(i,' ")"•■'■
This is zero, if q is different from both k and I; it is
= I ; and it i;
=fu:
Jc. V
?((,'*)«""•■'■
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362
if g = k. Thus
INFINITE SYSTEMS
[113.
U i)
= «ll,t^i + «(,(■»* ■
But all the quantities v^ vanish ; hence
We thus have
for all values of /; and ^ is any finite quantity, for only the
ratios of the quantities x are determiuabe.
Similarly, if D be of characteristic r, so that the minors of
lowest order which do not all vanish are of order r, let
he snch a minor different from zero. We then have
Thus the coefficient oi x
When q is equal to any one of the integers ^,, ^i, ..., 0r, this
coefficient is equal to a minor of order r — 1 and so vanishes.
When q is not equal to any one of those integers, the coefficient
is equal to a determinant with two columns the same, and it is
therefore evanescent. Hence
,S = 0,
and therefore
:::;:)"
T
A. .... /3„, A, /3„
■:::.>
where, on the righthand side, m must not be equal to any one of
the integers a„ ...,ar. It thus appears that there are r relations
among the quantities m; and that, in particular, each of the
quantities m.^, u,^, ..., ■«„,. is linearly expressible in terms of the
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113.] OK EQUATIONS 363
remamiiig cjiiantitiea u. Accordingly, we assume these r quanti
ties u omitted from consideration.
Denoting by a any integer other than a, a,, and by any
integer other than ^j, ..., ^r, we have
=(*:*::.■:;«:)'"'&",&;:;;:»:) '''(A.sT.r.'.A)''''
in the same way as for the simpler ease ; hence, as all the quanti
ties !t„ vanish, we have
U.A J"'.!, (a. A A._
SO that all the quantities xg a,rc linearly expressible in terms of r
such quantities.
For further properties of infinite determinants, reference may
be made to the memoirs quoted at the beginning of  109.
Ai'j'LicATioN TO Differential Equations.
114. When the differentiai equation is given in the form
the substitution
~l /'^•''^
W = we ■'
leads to an equation of order n in w, which is devoid of the term
involving r— ^ • The coefficients of the new equation are linearly
expressible in terms of Q^, Q,, ..., Qni, Qn, and the expressions
involve derivatives of Qj up to order n — 1 inclusive and integral
powers of Q,. We may therefore take the differential equation in
the form
^wi.^
' (/s"= "■ " ' de
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364 INFINITE DETERMINANTS APPLIED TO [1.14.
We assunie that, in the vicinity of 3 = 0, it possesses no synectic
integral, no regular integral, no normal integral, and no subnormal
integral. The point ^ = is then a singularity of the coefficients ;
and, if it be only an accidental singularity {of order higher than s
for Pg, in the case of some value or values of s), the conditions for
the existence of a normal integral or a subnormal integral are
not satisfied. We assume the coefficients P still to be uniform
functions of s, and we shall suppose that their singularities are
isolated points. Let an annulus, given by
Ji<\z\< R,
be such that its area is free from singularities, no assumption
being made as to the behaviour of the coefficients P within the
circle of radius R; then it is known* that each such coefficient
can be expanded in a Laurent series
P.= ic,,^^^ (r = 2, 3, ..., n),
which converges uniformly and unconditionally within the annulus.
Without loss of generality, it may be assumed that
Ji<l<li':
for, otherwise, we should take a new variable Z = e(RR')^, and
the limiting radii R and R' of the annulus for Z then satisfy
the conditions
R<l<R'.
Further, owing to the character of the convergence of P^, we
have
dP, 
d ( dPr\ ?
and so on; all these series converge uniformly and unconditionally
within the annulus. Hence also
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J 14.] LINEAR DIFFERENTIAL EQUATIONS 365
similarly converges within the annulus, where R (fi) is any poly
nomial in /J. ; and therefore, taking the circle I e\ = ]., every point
of which lies within the annulus, the series
_iiiWc,,„j
converges,
115. From the general investigations in Chapter ii, it follows
that the equation certainly possesses an integral of the form
J/=ZP<P (3),
where p is any one of the values of g— . logo), the qnantity w
being a root of the fundamental equation associated with an
irreducible (but otherwise simple) closed circuit in the annulus ;
and the quantity is a uniform function of z. As the integral is
not regular, the number of negative powers of 3 in is unlimited ;
and so we may write
In order to have an adequate expression of the integral, the
quantity p must be obtained ; the value of a„ h a,,, for m = + 1,
± 2, ..., + X , must be constructed; and the resulting series must
converge for values of s within the annulus.
We first consider the formal construction of the expression for
the integral. Let
<i>(p) = p(pl)...(pntl) + c,,_,p{pl)...(pn + S)
+ Cs,3p(pl)...(pn + 4) + ... + C„^,,n+iP + Cn,,,;
+(p + /i)...(p+>(m + 4)c,,r_^_5+ ...
...+(p+/J.) C„.,r«„+, + Cn.r^n ;
and write
Gm(p)=0(p + m)a™+rC^,^a^,
where, in the last summation, the values of fi are from — x to
+ CO, with /j. = m excepted. Then we have
p(,) = i G,(rt ..*,
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INTEGRAL TN THE FOEM OF
[115.
SO that y is an integral of the differential equation if
for all values of m from — oo to + <» , there being no assumption
that the negative infinity is the same numerically as the positive
infinity. Let
n
for all values of fi other thai
■^in.m with the convention
»)
ti; and introduce a quantity
ffm (p) = (m + p) S ^ni.
where the summation now is for all values of y:*. We then requi
the infinite determinant
"((>)[*.,,]
■  ■ T ''i''— 1 — y ) ^ » ^—1 * ^— I 1 ' '^—1 '
..., ^1,2 1 ''/'"l,— 3 ■ ''/'"!, II I 1 ' '"/''l,!
the necessary and sufficient condition of the convergence of which
is the convergence of the double series
for all values of m and fi between — co and + oo except m = f*.
116. In order to establish the convergence, we firat transform
the expression of Cm.,. Let
then we may take
(p+t^)(p+^ll)...(p+^lp + '\)
= (p + mX){p+m\l)...ip + m
= (p + my + «p,. (P + "O" + V. ip + '«)'
\p + \)
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116.J A LAURENT
where a,,,, is a polynomial in X of degree r. Using this for all the
terms in G^ „, we have
where
A,(X)
+A,{\)(p
+ «.). + ...
+ i,.W,
Accordingly, w
e have
^2S
<^ (p + m) 1
W,(x)j + SS
!*((> + »•) 1
!^.W! +
Now the series
'i" fl{i)o,.
+ S2t :
byk"
(Ml
converges for
every value
2, S, ..„ n
of r, where
! ii(\) is
any
polynomial in X. Hence
every term of which (for the various values of p) converges, because
««ii,sj!+s is a polynomial in X of degree s — p + 2, and therefore
the whole of the righthand side is a converging series. Accord
ingly, we may write
'F A,{X) = H„ (s = 2, ...,n),
and then each of the quantities ffs is finite.
We thus have
'.(p + mTjl ,(p + m)^
i!?t..,.<i^.i!rii^h™!ri^i
+ ...
"' . I * (p + m)
Assuming p to be any quantity, different from any of the roots of
any of the equations
*(p + m)0,
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368 INFINITE DETERMINANT [116.
each of which is of degree n, we know that all the series
converge absolutely, for the values k—2,S, .... n. Moreover, the
sum of each such series is a function of p : and then, if p varies in
a region no point of which is at an infinitesimal distance from any
of the roots of ^(p + m), the convergence of the series is uniform.
Accordingly, the double series
converges uniformly and unconditionally ; and therefore the infinite
determinant ii(p) converges uniformly and unconditionally, pro
vided p does not approach infinitesimally neai any root of any of
the equations 0(p + m) = O. Clearly, il(p) is a uniform function
of p, for such values of p.
Further, we ha.ve
^«.,.(p)0(p+m) = C,„,„(p),
and therefore
t^+,,^, ip)<f>(p + m + l) = t?^+,.«+, (p)
= .r,„,^(p + l)^(p + l+m),
so that
Construct the infinite determinant 0.(p + I), and then replace
each constituent ^m.^ip + 1) by ^,„+,^„+,(p); the result ia to give
the modification of il (p), which arises by moving eaoh column one
place to the right and by depressing each row one place, in other
words, by taking ^}r,^,(p) in the diagonal as the origin instead of
ifpo „(/)). But such a change makes no difference in a determinant
which converges absolutely ; we therefore have
n(p + i) = n(p),
or the infinite determinant fl is a periodic function of p.
Lastly, by making p infinitely large in such a manner, that it
does not approach infinitesimally near any of the roots of any of
the equations
.^(p + m) = 0.
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116.] MODIFIED 369
(which roots for different values of m differ only by real integers,
80 that if we take p=u + iv, where u and v are real, it will be
sufficient to take v large), we reduce to zero every constituent
that lies off the diagonal of il(p). As every constituent in the
diagonal is unity, and every constituent off the diagonal is zero, it
follows (from the law of expansion of an absolutely converging
determinant) that
Lim n (p) = 1,
provided p tends to its infinite value in the manner indicated.
Modification of the Infinite Determinant ii (p).
117. It is convenient also to consider another infinite determ
inant associated with fl (p). The equation G^ ip) = was taken
in the form
<l>(m + p)Xir^.^a^ = 0:
and the infinite determinant ii(p) was composed of the constituents
^m.f If ^"^ infinite determinant were composed of constituents
Ip {m + p) yJTm,^, then the row determined by the integer would
have a common factor 0(m4p); and thus there would be an
infinitude of factors, the product of which either should converge
or should be made to converge. Let pi, p^, .,., p„ be the roots of
(j) (p) = 0, so that
<f> (p) = (p  pi) (p  p„) . , . (p  p„),
and therefore
^.
To change this into a form suitable for an infinite convergin
product, we multiply by
with the convention
S.(rt1.
As A^(p) remains finite and is not zero for finite values of p, v
may replace the equation (?m(p) = by
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370 MODIFICATION OF THE [IIT.
Now let
for all valuPB of ?« except m = 0, and
Xo.'^ n (pp„);
also let
Then the equations between the constants a have the form
In association with these equatio
determinant
consider the infinite
(p)=
X
J
X^3
X>i
x».
X.
. Xl!. •■■
X1.2
XJ.1
X'V
X.
. X.,.. ••■
X.,.
X«,i
X",o
X...
, x... , ...
»,.
X>.i
»,.
Xi.i
. Xi.= . ■■■
Xt
%2,l
X.,.
X=,i
. Xs,B . ■■■
Taking the diagonal to be ..., X^.s, Xi.~" X'>.'" X'." %',!' ■■■• ™
require to establish, (i), the convergence of the series
summed for all values of m and /*, except m = fi, from  ;» to + k
and (ii), the convergence of the series
S(x,.l),
in order to know that the infinite detenninant D (p) converges.
We consider first the double series %'Zxm,u ^s*
i,(p) = .n*„(f).n.
(m + 0).
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117.] INFINITE DETEHMINANT 371
The quantities p^, pi, ■■, pn are finite; hence, so long as p remains
within a finite region that does not lie at infinity, there is & finite
quantity K which is larger than any value of j/'m(p) for values of
p within that region. Hence, as
.\<K'
when m is not zero. When m is zero, we have
»,,=*.WC.,,o..,.
Proceeding exactly as with tlie series SS^m,^. in § 116, summing
for all values of m other than zero, and for all values of fi other
than m = /i, we have
lf + "
+fflX
lp + »
■\tl.\t
every term of which is finite, and therefore
is finite. Also
l\(!..A^\HJ\p'\ + \H.\\p''\ +
which is finite, so that
converges. Hence the double series
Slimmed for all values of m and /t between — co and + c
, converges.
Moreover, all the series, which
superior limits in the inequalities, converge uniformly withir
region of p considered ; hence the double series converges
formly.
The establishment of the convergence of the series
i(fc,.i)
except
in the
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372 CONVEEGENCE OF [117.
is simple. We know, by Weierstrass's theorem*, that tho series
converges uniformly and unconditionally; so that, if
»„,,. = x.,i.
the infinite product
n(i + (?™,™)
converges uniformly and unconditionally; and therefore^
n_(i + 9.,,)
converges. But
n(l + ^™,,„)<l+2^,„,,„;
hence S ^m,inl converges uniformly, that is, the series
_S(x,.,«l)
converges uniformly and unconditionally.
The eoiivergonce can also be established as follows. Let
=(^').
and choose a finite positive integer p, such that, for values of p under c
sideration, we have
\pp'\<P,
where p' is any one of quantities pi, p2, ..., fn The sum of the terms
J(x«.I)
is finite, and may be omitted without affecting the convergence: and we e
sider the sum of the remaining tenns, for which we have
\m]>p.
We have
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117.]
THE DETERMINANT
and therefore
2>ii^7'^<
<
.j / 1 p 
f' + 
Pff^
\pi'.r\^
»? " ^
1 \9pA
Now for all the values of m under consideration
and therefore
J pP"[
^1 ipP'L,
 p+l p+l'
^.K^P^
P,\\
so that we maj
'take
'■S"
?^)\
where
1f.<i.
Hence
Xm.
„=nM^
ttj 2,,{,,,l"
Now 2 (iir(p—p^)2 ia finite for all the values of puuder conaidenttion, and
it is finite for all values of m if ji^ involves m; let ^denote the gieateat value
of its modulus. Again, for any quantity 0, we have
othat
r writing
A' jV! 1 iV=i ]
<i4'(4)"4:(4.)^
shewing that the si
converges.
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374 EVALUATION OF THE [117.
The infinite determinant I)(p) thus converges uniformly and
unconditionally for all values of p in the finite part of its plane.
Its relation to li (p), which converges similarly for values of p
that are not infinitesimally near any of the roots of any of the
equations ^{p + m) = 0, is at once derivable from its mode of con
struction from n(p). The row of quantities Xnt.j'lp) ^^ ^(p) f*^^
the same value of m is derived from the row of quantities i^m.^ in
il (p) for that value of m, through multiplication of the latter by
h^(p)(f>(m + p).
Hence
D(p)a(p)nj,.(p)^(„,+p)
where
ii(p)=nK(p)<l{^ + p)
Jt [i{(' * 'if"') " " '""']] .i <" " "•'•
and 11' implies multiplication for all values of m between + oo and
— cc except m = 0. Also
n(/')=^'^j;Msm(pp.)'rl.
Now D{p) has been proved to be finite (that is, to be not
infinite) for all finite values of p ; and manifestly, from its form, it
is a uniform funct.ion of p, so that it is a holomorphic function of
p everywhere in the finite part of the plane. Further, D,{p) is a
uniform function ofp; and it has been proved to be not infinite
for values of p, which are not infinitesimally near any one of the
roots of any of the equations (p(p + m) = 0, the aggregate of all
these roots being
pi + m, p^im, .... pn + 'm, (i« = — oo to + oo ).
Hence, owing to the relation
D{p)n{p)n(,p).
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117.] DETERMINANT 375
it follows that these roots are poles of ^(p) Take a line in the
pplane inclined at a finite angle to the axis of real quantities,
choosing the inclination so that it does not pass thi'ough any of
the points p^ + m for all values of <r and m ; let it cut the axis of
real quantities in a point f. Take the point /41 on that axis,
and through it draw a line parallel to the former, thus selecting
an infinite strip ia the /jplane. Since
a{p + i)^a(p),
the uniform function il{p) undergoes all its variations in that
strip: and within the strip, we have
Lim fi (p) = 1.
Owing to the nature of the poles of il (p), the strip contains n of
them, which may be regarded as the irreducible poles : suppose
that they are pi, p^, ...,/3«. Within the strip, p= oo is an ordinary
point of the simpiyperiodic function ii (p) ; it follows* that the
number of its irreducible zeros is also n, account of possible
niuitiplicity being taken ; let these be /j/, />/, ..., p«'. Hence
n(p) = A ^^^ t(P  Pi') '^j si n {(P  PaO ' ^1  ■ ■ sin {( p  Pn) ■^]
sin i(p  pi ) wj sin Kp  p, ) tt) ... sin Kp  p„ ) tt) '
taking account of the holomorphic character of D(p) for finite
values of p, and of the relation
Here, A is independent of p. To determine A, we use the
property
Liran(p) = l,
which holds for
p = u+ iv,
in the Hmit when v is infinite, whether positive or negative.
Taking v positive and infinite, we have
and taking v negative and infinite, we have
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376 FORMATION OF [117.
Hence XpJ — Sp„ is an integer ; if it is not zero, we can make it
zero by substituting, for the quantities p',, values congruoiit with
them. Assuming this done, we have
2;>/ = Sp„,
^ = 1,
so that
ilip)'
and therefore
Moreover, the quantities pt, pi, ■■■, pn a^e the roots ol ^{p)~ 0, ao
that
hence
Sp.'i»(«l).
118. Next, we consider the expression
»in((p
Pi.
T
...sin((p
Pn
)■']
sinUp
Pi"
)ir\
...sin(((>
■Pn
M'
DM
: n
.h.
iKpji'W
=IQ^
i to prove that this series converges for all values of z
within the annulus. It manifestly arises from D (p), on replacing
;\;o fc in D (p) by 3* ; we shall therefore assume that F is transformed
into this modified shape of i)(p). When the determinant is in
this shape, we multiply the column associated with m by z~™, and
the row associated with m by z™] these operations, combined, do
not change %m,ra, and they do not alter the value of the determ
inant. Let this combined pair of operations be carried out for
all the values of m from —x to + co ; the result is to give a
determinant, which is equal to Y and has
Xv,,"~'
for its constituent in the same place that ;^j,^, occupies in 1) {p).
Hence, as for D (p), so T converges uniformly and uncondition
ally for values of p within the pregion selected, and uniformly
and unconditionally for values of z within the annuius, if the
doubly in finite series
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118.] AN INTEOBAL 377
converges uniformly and unconditionally within those regions,
and if
S(x«1)
converges uniformly and unconditionally.
The latter condition is known to be satisfied, owing fco the
convergence of J) (p). It remains therefore to consider the con
vergence of the double series.
With the notation of §§ 115 — 117, we have
Now
A^ (\) s' = a;„_^,. C;, x_5 3^ + a„_3, ,_, c,, x_, j^ + ...+ Cr. w sh
owing to the definition of the coefficients in the original differen
tial equation, the series
converges uniformly and unconditionally, for values of a within the
annul us
R<\2\<B';
and therefore the series
converges uniibrmly and unconditionally for the same range.
Denoting this by J^, we have
and \Jf\ is not infinite for any of the values of z.
Again, as (§ 117)
and
when m is not zero, we have
■*.W%r^^
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378 CONSTRUCTION OF [118.
Proceeding with the double series SSxni.M^'""^ exactly as with
the double series £x"i.f > omitting for the preaent the terms corre
sponding to m = 0, and remembering that the summation is for all
values of m other than m =/i, we have
<K^J
every group of terms in which is finite, so that
SSx,,,^»
is finita Also, taking account of the terms omitted for the value
m — 0, we have
S c.,,^' < \J,: If'", + i./. p"; + ... + IJJ,
which is finite, so that
summed for all values of m and ^ between — oc and + co except
m = /i, converges unconditionally. Moreover, all the series which
occur in the supenor hmits in the inequalities converge uniformly,
both for the values of s considered and the retained range of p ;
hence the double series converges uniformly and unconditionally.
The proposition is therefore established for
£©"■
A similar investigation shews that the series
for any value of r, the numbers a and /S being any whatever,
converges uniformly and unconditionally for values of z within
the annulus, and for values of p in the range that has been
retained.
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119.]
CONSTBUCTION OF IrREGULAE INTEGRALS.
119. These results may now be used, by a generalisation of
the method of Frobenius in Chapter iii, to construct expressions
for the integrals of the equation
Writing
1/= 2 (I™ 3''+'^,
and adopting the notation of § 115, we have
e,(p) «»+'»,
it GmWO,
for all values of m between — oo and + oo , except m—i. The last
equations are equivalent to
/i„<p)e»(p)o,
that is, to
for all the values 0, + 1, + 2, ... of m, except m, = i. Let
We have
that is,
for all the values of h. Hence, writing
o...Af'V
we have
and
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380 IRREGULAR [119.
Thus the quantity y, where
^^!©^'"
satisfies the equation
The determinant D (p) is of normal form ; the series for y con
verges uniformly and unconditionally, alike for vahica of s within
the annulus R<\z\<R', and for values of p within the finite
region contemplated.
120. Let p=p' be an irreducible simple root of D(/j) = 0.
Then the first minors of constituents in any line cannot vanish
simultaneously for p — p'', for
and the lefthand side does not vanish for p = p'. Selecting minors
of constituents in the line i, we havo
and
that is, ^1 13 an integral of the equation.
Similarly for any other irreducible simple root of D {p) — 0.
121. Next, let p = p' be an ineducible multiple root of
i> (p) = of multiphcity a.
Firstly, suppose that some of the first minors of D(p) do not
vanish for p = p'', let some of these nonvanishing minors be
minors of constituents in the line i. Then, in the vicinity of
p = p, we have
as a quantity satisfying the equation
P (2/) = ^sMi" (p  pf R(p p'%
where Rip — p") does not vanish when p — p. It therefore follows
that
^^ipp'y'^^i^'P'P')'
yGoosle
121.] INTEGRALS
SO that, if ^< 0" — 1, we have
\dp''),~/
is an integral of the equation. Hence, corresponding to the
irreducible root p' of multiplicity a, there are integrals
^'^^\h 0] ^'^'' "^ ^" ^"^ ^ = ''^ + ?/« i«g ^^
3/. = 2 r^, Ql zf^" + 2^, log ^ + y, (log s)=
= 9jj + 2iji log s + y, (log ^)^
y..^7),_i + (^l)7,,„,log^ + ^ ''~y^^^ "^^^.^(log^)' + ...
... +(<r l)j?,(log3)^' + yoaog^)'"'.
when, in each of these expressions on the righthand side, we take
p = p.
122. Next, still taking p = p' to be an irreducible root of
i> (p) = of multiplicity a, suppose that, of the minors of successive
orders, those of order r are the first set which do not all vanish
for p — p'. Let the lowest multiplicity of p' for first minors be o,,
for second minora be o^, and so on up to minors of order r — X, the
lowest multiplicity for which is denoted by a,,. Then, owing to
the composition of B in relation to first minors, to the composition
of first minors in relation to second minors, and so on, we have
<r><ri>(r,>...>o^,.
There are two ways of proceeding, according as r < ff, or r = a.
First, let r < cr. With the preceding notation, wo have
and
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382 SUBGROUPS OF [122.
After the explanations given in the construction of these expres
sions, we know that p= p is a root of m.ultiplicity oi for some
of the minors in the expression for y. As before, in § 121, the
quantities
dy 3^
^' dp' ■■•■ dp''
when in each of these we take p — p', are such that
for A, = 0, 1, ..., <r — 1. But owing to the fact that p = p' ys, a root
of all the minnis I ,1 of multiplicity ctj, all the quantities
dy d^y
■'' dp' ■■■■ dp''
vanish when p — p' Hence the n on evanescent integrals which
survive are
dp"' ' 3p''i+^ ' ""' 3,3'^' '
when p = p'. They have the form
J,,. = aI^ Q «'« + (a, + 2) ,„ log .
and so on : their number being
Next, p — p' is, a root of least multiplicity oi for some of the
minors of the constituents of any line i: and there mast be at
least two such minors. For
iwsQv
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122.] IRREGULAR INTEQBALR §83
if p = p' 13 & root of multiplicity oj + 1 for all the minors but
[ , ] , then, as it is of multiplicity ^ .Ti + 1 for Z* (p), it would be of
multiplicity o"! + 1 for f J. Similarly for any other line. Once
more substituting
in P(y), we have
= t?i (p) z'+'" + Gj (p) z<+^^,
provided G^ {p) — 0,
for all integer values of p from co to +x except p — i, p=j.
The last equations are equivalent to
kp(p}G^(p)=0,
that is, to
for all integer values of p except t and j.
Consider quantities ag of the form
for ail values of 0, tho quantities A and B being arbitrary. With
these expressions for a^, we have
Each of the sums on the righthand sides vanishes, when p is not
equal to either i or j : and thus the preceding expressions satisfy
the equations
«,((>) e,(p)o,
for all integer values of p except i and j. Further,
*,(p) ft w ='?»,,«,
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384 SUBGROUPS OF [122.
and, similarly,
i,(rte,(rt=^(;)+i)Q.
Using these values, we have
as the expression for y ; and it satisfies the relation
i* iy) = Gi (p) ^"^"" + Gj (p) 3^+j»
As the righthand side of the last equation has p^p' as a root of
multiplicity aj, the quantities hi(p) and hj(p} having no zero for
finite values of p, it follows that
(PI.'
for X = 0, 1, ..., ffj — 1. Therefore all the quantities
dy d"''y
when p = p', satisfy the equation P (w)  0. Owing to the form of
y above obtained, which has p = p' rr a root of multiplicity <rj, all
the quantities
^' dp' ■■■' dp"''
vanish when p — p. Therefore the surviving in1
'"if
,d so on : their number being
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122.] IRREGULAR INTEGRALS 385
Similarly for the next subgroup. With the same notation as
before, we have
P (y)  G, ((>) ^'+'" + B,W «'+'■ + a. (p) ^■+»",
provided
for all values of^, other than i,j, h, from — « to « . The analogy
of the preceding case suggests
for all values of 6, where A, B, C are any quantities. With these
expressions for oe, we have
=AX
Each of the three sums on the righthand side vanishes, when p is
not equal to either i or j or k: so that the preceding expressions
for a# satisfy the equation
e,(/>)o,
for all values of p other than i or j or h. Further,
Thus
where ^(e, p) is a linear combination of minors of the second
order; and
the coefficients a^ being linear combinations of
third order.
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386 SUBGROUPS [122.
As *(2, p) has p=p' as a root of multiplicity a^, it follows
that
for X = 0, 1, ..., C72  1 ; HO that all the quantities
^' dp ■■■' ap"'
when p — p', satisfy the equation P (w) = 0. Owing to the form of
the coefficients a^ in y, each of which has p^p' as a root of
multiplicity a^, all the quantities
dy d"_^y
^' dp' ■■■' dp"''
vanish when p~p' \ and we therefore are left with the integials
and so on : their luiniher being
Proceeding in this manner, we obtain successive subgroups of
integrals ; the total number in the whole group is
which ia the multiplicity oi p = p' as & root of D (p) = 0.
123. Two cases, both limiting, call for special mention.
It is manifest that, if tr — oi > 1, the first subgroup contains
integrals whose expressions involve logarithms; likewise for the
second subgroup, if ffi — (ra>l; and so on. If, then, all the
integrals belonging to the multiple root p = p' of Z* (p) = are to
be free from logarithms, we must have
0 01 = 1, ojo,1, ....
yGoosle
123.] OF INTEGRALS 387
and there fort!
which thus is a limiting case of the preceding investigation.
An intimation was given that, when r = o", a ditiferent method
of proceeding is possible. As a matter of fact, the property of the
infinite system of lineHi relations, ostablished in § 113, leads at
once to the result. Let
be one of the nonvanishing minors of order r belonging to D (p) ;
then
and the quantities a.^^, a^^, ,.., a^^ are bound by no relations, so
that they are arbitrary constants. The integral determined by
these coefficients is
it manifestly is a linear combination, with arbitrary coefficients
«■>.,, •■■ C'Mr, oi r integrals which are, in fact, the group of integrals
above indicated.
The other limiting o;\se occurs when r = 1 : all the <r integrals
belong to a single subgroup. In that case, there exists at least
one minor of the first order which does not vanish when p=p';
the condition is both necessary and sufficient.
124. We thus have a set of rr integrals, belonging to an
irreducible root p' of D(p) = which is of multiplicity o.
Similarly for any other irreducible root of D{p) = 0; hence, when
all the irreducible roots are taken, we have a system of n integrals.
We proceed to prove that this system of integrals is fundamental.
For, in the first place, it follows (from the lemma in § 27) that
the integrals in any subgroup are linearly independent, on
account of the powers of log^ which they contain.
Next, there can be no relation of the form
Ci2/.. 1 + O^y^ , + ... + C^yr.i = 0,
yGoosle
388 FUNDAMENTAL SYSTEM [124.
with non vanishing coefficients 0. If such an one could exist, the
coefficient of every power of z in the aggregate expression on the
lefthand side must vanish. Writing
i, j, h, ... =Pi, Pi, Pi, ■■, Pr\
k, I, m, ... =5,, q„ q„ ..., q^ '
we have
"M'e.
'PuP2,;pA , (Pl,p2,
'P„P^,
and the quantities Ag,, A^^„ ..., j4s.s f're at our disposal. Let
tiiese last be chosen so that
Then the coefficient of sf'+'i in ^s,i is zero if (<s, and it is different
from zero if i = s: let it be denoted by [ye,,\
The above relation being supposed to hold, select the co
efficients of 3^'+^', 2*'+?', .... z*''+9r in turn. As they vanish, we have
0. b.,i + c. [*.]., + ... + c, [».,]„ = 0,
from the coeSicient of sf'+^^ ; every terra vanishes except the first,
and [yi.i\, is not zero ; hence
Cj = 0.
The vanishing of the coefficient of if'*^' then gives
0. [.»., 4, + c. b.,.],. + . . . + C, [,j,, ,],_ = ;
every term after the first vanishes, and [ya.ijg, does not vanish;
hence
And so on ; every one of the coefficients G vanishes ; and thus no
relation of the form
yGoosle
124.] OF INTBGEAI^ 389
Next, there can be no linear relation among the o membeis of
a group. For, in any expression
SO,,j,„
the coefficient of the highest power of log z is of the form
and this can vanish, only if the coefficients Gs_i are evanescent;
hence 'ZG^tys.t can vanish, only if the cocfiicients Gs_t ai'e evan
escent.
Lastly, there can be no linear relation among the members of
different groups. For let Y{p',z), Y{p",z),... denote the most
general integrals of the groups belonging to the irreducible roots
p', p", . . . respectively, of D (p) = 0. Let z describe a contour
enclosing the origin; then Y(p', s) acquires a factor e'""^, Y(p", z)
acquires a foctor e^^", and so on. Thus, if there were a relation
aY{p\s)+0Y(p',z) + ... = Q,
then
ae«">' Y(fi', z) + ^e="*" Yip", z)+...^0;
and similarly, after k descriptions of the contour,
ae^ip'' Y {p, z) + /Se=^^"" Y (p", s) + . . . = 0,
for as many values of the integer k as we please. Now p', p", ...
are the irreducible roots of D (p) = ; no two of them are equal,
and no two can differ by an integer. Hence the preceding rela
tions can be satisfied, only if
in other words, no linear relation among the n integrals can exist.
They therefore form a fundamental system.
The Equation D(p) = is the Fundamental Equation of
THE Singularity.
125. Consider the effect which the description of a closed
contour, round the origin and lying wholly in the annulus,
exercises upon this fundamental system. Let
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390 FUNDAMENTAL EQUATION [125.
and let y' denote, at the completion of the contour, the value of
the integral which initially is y. We have
and so r
where
Hence the fun(! a mental equation {Chap, ii) is @ = 0,
&~e.
,
, ■
,
a.i ,
efe.
, .
., .
a^i ,
e'e, .
,
,
,
, .
., A, ,
0,0,.
«'», 0, .
, ...
., , ..
■ , ,
.. ,
0,0,.
0,0,.
«, ..
where a' is the number of integrals in the group belonging to the
root p of D{p)=0 of multiplicity a' ; a" is the number in the
group belonging to the root p" ; and so on.
Now it was proved that
Dip).
n I
■Kcp.')"!.
if
Henof
sin (p  p.') ,r = g «  1' + '.') " (.2">  e''
^^1"^  (ZSji'""'"""^''','?/''" '^'■
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125.] OF THE SINGULARITY
Also (§ 117)
lp„' = in(nl),
so that
gT'2p„'=,gin(«l)Ti^ + 1;
and therefore
Dip)
As the quantity e""*"^' has no zero for finite values of p, it thus
appears that, so far as roots are concerned, 2> (p) = and = are
effectively the same equation, when the relation between p and d
is taken into account. Also, so far as roots are concerned, il{p)=0
ia effectively the same as D(p)=0; hence ant/ one of the three
equations
= 0, D(p) = Q. n(p) = 0,
may be used for the determination of 6 and the associated
quantity p.
It is known that = is ao equation remaining invariaative
for all raodiiications of the fundamental system : and, for the form
of equation adopted in § 114, the term in independent of is
equal to unity (§ 14), This property in the present cose is verified
by means of the values of the quantities 8', 0", ...; for
(_^y(_^Y'... = (l)''e^'''^''"' = (lf e"l"^J^' = ( 1)".
The remaining coefficients in are known (§ 14) to be the in
variarUs of the equation, whatever fundamental system be chosen.
Replacing by ii (p) for purposes of this discussion, we have
^ si n {(p  p/) tt] ... sin {(p  p/) 7r
^^' sin(pp,)7rl...sin(pp„),r!
I^ow
where
so that, as
and therefore
sinl(pp,)»l »«,■
ipr  ip;,
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392 FUNDAMENTAL [125.
we have
^P' (<)e,)(e8,)...(0~en)
^_. .. + (_!)«■
Hence, when il(p) is expanded in descending powers of d, the
term in 8" is unity ; and when it is expanded in ascending powers
of 6, the term in 0" is likewise unity.
When the quantities &,, 0^, ..., 6^ are unequal, then Xi(p) can
be expressed in the form
On account of the character of il (p), when expanded in ascending
powers of 8, we have
so that there are w — 1 independent quantities Jl/,', and these are
equivalent to the w  1 invariants. The equation may also be
expressed in the form
n{p) = l + t M, cot [{p  p„) tt],
where
and therefore
i #,= 0.
Corresponding expansions occur in the case when equalities
occur among the quantities p^, p^, ..., pn
126. The integrals, which have been obtained, are valid within
the annulus represented by R^\s\^R'; the inner circle may
enclose any number of singularities of the equation, and the outer
circle may exclude any number of other singularities of the equa
tion. But care must be exercised in particular cases. If for
instance, the only singularity within the inner circle is the origin,
and the integrals are regular in the vicinity of the origin, then in
the expression of any integral, such as
yGoosle
126.] EQUATION 393
there can be only a finite number of terms with negative values
of m : the method, which is baaed upon the supposed existence of
an unlimited number of such terms, is no longer applicable. If
the only singularity outside the outer circle is z — <xi, and if the
integrals are regular in the vicinity of 2 = cc , then in the expres
sion of any integral, such as
there can be only a finite number of terms with positive values of
m. : the method again ceases to be applicable.
In 3u<ih cases, the best procedure ia to construct a fundamental
system which shall include the regular integrals : this ia the
customary procedure for, e.g., Bessel's equation, the integrals of
whicli have 3= co for an essential singularity and are regular near
s = 0. The method, which uses infinite determinants, is best
reserved for equations which have their integrals nonregular in
the vicinity of every singularity : it is nugatory when applied to
Bessel's equation.
Ex. 1. Consider the equatio
S+S+ b"
It is clear that the point 3 = is an esBential singularity, there being i
integral regular in its vicinity, when a is different from 0: and that 2=co
likewise an essential singularity, when y is different from 0. "We ahall aasun
that both o and y arc non vanishing quantities.
Let
the eqiiation becomes
dhi (a h a\
With the notation of the preceding paragraphs, we have
4.(p)=p(pl)l6=(pp,)(ppj);
Or,g,=(i, when /t<!' 1, and when fi> r+l ;
^,,^=0, when /i<r 1, and when >i>»+I.
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394 EXAMPLES
Henco the value of n (p) is
[126.
..., 0,
*(p""2)
1
' *(p2
f
"
,
, 0,...
..., 0,
0(pl
'
' f(pi)
<>
,0,,..
,.., 0,
' Hp)
1 1
w
, 0,...
, 0,
^(H^i)
1
' *(P + 1)
, 0,,..
..., 0,
"
LI
,
*(p + 2
'
*(7+2)' "'■■■
The general investigation ahewa that, wheu p, and p, are unequal (which
will he assumed),
Q(p) = l+ifi7rcot{ppi)n + J/3^cot{ppj)jr,
with the condition
that is, we have
Q(p)=l + 7rJ/[coti(pp,)ff}Mt{(pp2),r}],
where M is itidepeiidcnt of p.
Taking the determinantal form for Q (p), and espanding according to the
law established in § 110, we have
where odd powers of a do not occur because the combinations which they
multiply all vanish. Also
, ™+i0{p+m)0(p+m.+l)^(p+?))<^{()+F+l)'
2 2
^(p + m)>(p + ™ + l)^(p+p)<^(p+p+l)<fi(p+g)0(^ + 5 + l)'
Hence we have
To find jVj, we notice that the only terms in M^, which have p = pi for a
pole, are those given by m=0, m=  1, these being
<^(»^(p+l) 0(pl)^(p)"
yGoosle
126.]
hence
<j>{p)={l>p,){pp2)\
P,pA<Pipi+V 0(pii)J
PiPa lipip^y ^^ipipi)
Again, writing
$(p + m) = ^(p + ra)(>(p + Mi + l),
we have
^*=^^*{p + m)*{p+p)
Consider
ij_„*(p+«)}'
it contains the terms
ik^J"
which do not oc
;cur in i/^,; it contvins terras
which do not oc
icur in J/, ; and it contains the terras
1
^>«+i*{p + '«)*(p+p)
twice over, onci
.in the form
and once iTi the form
I
Hence
»>^+i$(p + m)*(p+^)'
{?„*(p+7t)) JA*{p + r>:)i '.".*(p + ™)*{p4
»n,*^''.
80 that
^*=*{1I
t{p+n)\ ^„J'. l*{p + «)) „*{p + ™.
1
)1.(p+m + l)
The first term ■
on the righthand side is
= i.V3i'^2[cot(ppi)^^COt(pp,),r]S;
the residue of this function for p = p, is
= lS\^,vQat(p^p^)^
7rC0t{(pjpj)«}
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396 EXAMPLES [126.
In the second tci'ni on the righthand side, ttie residue for p=pi can arise only
for the values n = 0,n= — I; thus it is
, (l+2p2pa)(2 + p,p,) , (l2pi + 2pa)(2Pi + P3 )
' {PiPi)Hi+PiPif ^ (PiP3)={lPi + P2)'
after reduction. Similarly, from the third term, the residue is
38?> l8i
86«(3 + 4i)(p,p,) Sbmih)(p,p,)
Hence
.r^_w_cot{(pip2)^)_ 3H6526'^ + lfii3
* 1662(146) " 86^{3 + 46)(l4i}(pip5)'
after reduction.
Other coefficients could be calculated in a similar manner : but it is clear
that even iVg would involve considerable numerical calculations, and it is
difficult to see how the general term could thus be obtaiued. But the
method of approsimation may be effective in particular applications. Thiis,
in Hill's discussion* of the motion of the lunar perigee, the convergence is
very rapid; and comparatively few terms need be taken in order to obtain
an approximation of advanced accuracy. When this is the case, the values
of p' for the integrals arc given by
Q(p)=0,
cos2p^=cos{(p.p,)^}2^Jfsin{(pips)^h
and two irreducible values of p chosen are to be such that
Pi'+P2'=piIP2 = l
The expressions for the integrals are to be obtained. Denoting still by p
either of the quantities p,' and pj', the relations between the coefficients are
*oS:7)"'>+°'+.),(,T^'^*'°
and considering in particular the row 0, we know that the constants a are
propoi'tiona! to the minors of the constituents in that row iu the determinant
n (p). Thus
for ail positive and negative values of k : so that, if we take
imoir already quoted ii
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126.]
and our solution h.
for the effective expression of which, it is sufBcient to find the first minors, as
the series is known to converge within the annulus.
I« «rd» to obt.d„ Q torn aO.), we npl,«. ^^^, ^"^^y ^j^^
by zeros ; it will therefore be necessary to do this in the e
We ttus have
Q=^l + a'M^,,+ a'M^, + ...,
P+J:+S_
<p(p)'p(_p + l) 0(pl)0(p)
Sinailarly for M^ f from M^ ; and so on.
In order to obtain [ , j from ii (p), we replace — — ^ in the — 1 column
\  V 9 ip)
by unity; the quantities 1 and jf—~i,\ in that column by zeros; and the
quantities 1 and ■ . in the line by zero. Wo then easily find
\ij 4>{pi)
'"'^^(p)<t,{p+i)^<!>{pi)<i,(py<t.{pi)<i>{p2)
<p{pl)4<{p2}
aud .similai'ly for the others.
In the same way, we have
(i)*iVi)+*"«*^'..'+
^■f.(p + l)^(p + 2)'
and so for the others.
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398 MODES OF CONSTRUCTING [126.
Lastly, for iiogfttive values of p Icaa than  1 and for positive values
gieater than ll, we have
{^\==a^M^,,^\a^M„,^V ..
where
M, = Vn , H — — H
and iu fti the tlei'!
Alter the remarks made m lalatioa to the foimal development of Q (p)
it 19 manifest thnt these expressitna for the integrals are mainly useful foi
apprciimate numerical exparsions they cannot it ptesent lo held ti
constitute a complete formulation of the mtegiib
El 2 In his classical memoir alieidj quote 1 H 11 c3naiic.n> tie
h fli h Kg US erab mailer than «, The mi^inoir
WW ensaiof ean (jCLOimt bemg takpn of the
la una, nrgn pLd yPu'ar) and for the numern,il
approxima ns
It will bo noticed that the eflectiveness of the method is lai^ly influenced
by the data as to the smallness of a^, o^, .... when compared with a^.
Ex. 3. Disciaa the equation in Ex. I, when 6 = J, so that px — H
Ex. 4. Given an infinite system of differential equations of the form
lit
:J^a^,„:r„, (™^1, 2, ..„«>},
where the coefGcients o^^ are regular functions of t within a region \t\ ^ R,
such that ]«„,„! <S™,d„ in this region, where S^, J„ (for ra, w=l, ...,<») are
such that the series S,^, + S2^24...+(S„j1„ + ... converges. Shew that, if a
set of Constanta e^, Cj, ... be chosen, so that the series
Cl.d, + C2.d2+...+C„^„+...
converges absolutely, then a system of integrals of the equations is uniquely
determined by the cnndition that a',„ = (^,„, when ( = 0, for all values of ra.
(von Koch.)
Other Modes of constructing the Fundamental Equation
FOR Irregular Integrals.
127. The preceding nnethod, so far as it is completed, leads
to the determination of the fundamental equation for a closed
circuit round the origin, the circuit lying entirely in the annulus;
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127.J THE FUNDAMENTAL EQUATION 399
and it leads also to the determination of the integrals. Other
methods have been proposed by Fuchs*, Hamburger ■(■, Poincar^,
and MittagLeffler, some of them referring solely to the con
struction of the fundamental equation. But all of them seem less
direct than the preceding method, due to Hill and von Koch ; and
they are not less devoid of difficulties in the construction of the
complete formal expression of the integrals.
Ex, \. A modification of Hambui^er's method, applied to the equation
abeady discussed in Ex. 1, § 126, may give some indication of his process.
Changing the variable from ^ io t, where
the equation!] i°'' ^ ''
where
df
c=6i.
Let X describe a circle round the origin, say of radius unity ; then on the
completion of the circle, t has increased its value by 2ir.
Let y=f{^), y=3 (a^) be two linearly independent integrals ; and when x
describes its circle, let these become [/(if)], \ff (^)], respectively, so tliat
[/(^)] = «,./(^)+<tisS'W,
[6'(^)] = %/(^)+«22ffW
The fundamental equation for the circuit is
* CrelU, t. Lxsv (1873), pp. 177—223.
+ Crelle, t. Lissin (1877), pp. 185—209. In oouneotion with this r
reference shoulil be made to two papers by Giinther, Crelle, t. cvi (1890), p[
336, ib.. t. cvri (1891), pp. 298—318.
X Acta Math., t. iv (1884), pp. 201—319. In connection with tliis i
reference should be made to Vogt, Ann. de VEc. Norm., SSr. 3% t. vi (1689),
pp. 3—71.
5 Ada Math., t. sv (1891), pp. 1—33.
II In this form, it is a spcoial case of Hill's equation : nee Es. 2, g 126.
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400 EXAMPLE [127.
that is, by Poincare's theorem {§ 14),
<"^(l'll+«22)'» + l=0,
so that Ci, +<i22 is the one invariant for the circuit.
Let
The fundamental equation is indepondejit of the choice of the linearly
independent system, and it is unchanged when any particular selection is
made. Accordingly, let the integrals be chosen so that
F(i) = l, F'{t)=0, G{t) = 0, O'Wl,
when ( = 0; then, using the foregoing equations, we have
/■(2,).«,„ ff'(2,).o„;
and therefore
which accordingly gives the value of the invariant, when the values of i''(27r)
and G' (27r) are known.
To obtain these, let
so that a increases from to 1, as ( increases from to Stt. The equation
becomes
and this remains unaltered when we change u into 1  u. Two linearly
independent integrals, constituting a fundamental system in the vicinity
of M = 0, are given by
where a,=l, Cii=l ; also «„ is the value of 6„ when p=0, and c„ is the value
of 6„ whet! p=i, the quantities 6„ being given by the equations
(p + 2)(p + f)6a = !{p+l)2 + 4i^ + 8a}&,64a,
and, for values of )t^3,
()i + p)(!i + pi)6„={(n + (>l)'H4c + 8ai6„_i64(i6,^j + 04o6„_3.
Similarly, a fundamental system in the vicinity of u~\ is given by
Z, 2 «„(!«)", Z,= S c„(l«)"H.
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127.] EXAMPLE 401
Tho integral F{i), defined by the initial conditions
/"(() = !, F'{t)^0,
wlien i=0, is given by
The integral f?[i), defined by the initial conditions
i3(0 = 0. ^'[0 = 1.
when t — 0, is ^b/^n by
(;(<) = 4F2.
To obtain expressions for Fi^n), Q' {'in), consider values of u, which
lie in the vicinity of m=1 and are less than 1, By the ordinary theory of
linear equations, we have
First, let «=, so that 1 — « = ^; then we have
F{':r) = AF{^)+lBQ{n), a{^) = iCF{w)+DG{n).
Next, differentiate with regard to a, and then take m=J, 1 — a=J; we have
F {^)^  AF' MiBG' {t^), 0'{n)=^CF{n)DO'{n).
Moreover,
F(()G"(()F'(0<?(0=constant
= 1,
by taking the initial values ; hence
These rel^ttions give
A~FM e'M+?"M (?(,). A
Hence
e(2flT) = 4r2(2nT)
F{2,t~T)=Y^i2nT)
.AZ^{r)*BZ,{T),
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402 EXAMPLES [127.
Now, when t is tt, the value of u is ^, so that
and therefore
■.„+«,.f'(2,)e'(!,)
— L.i!".i. 2" .:.2— .ioH
whieh 13 the inyariant of the fmidimeutil equation This gives a fiimal
expreosion, the onU operatuno requiied lieing m the direct construction
of i^ and c„, ■ind no one ot thene tperation"; k inverse but the result
IS less fcuited to numem il ippiiiimation than la the method of infinite
determinants in the case when a h small
"We ah^ll return late! (§§ 137 130) to t difterent diSCusBion of this
equation.
Ex. 2. Applj the preoeditig method to Hill's equation
 j^=ao+«iC08 2(+aaCos4(+...,
in the case when %, a^, ... are not small compared with o^.
Ea;. 3. Discuss, in the same manner as in Ex. 1, the equation
In particular, obtain espressions for the invariants of the fundamentiil eqiuv
tion for z=0.
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CHAPTEB IX.
Equations with Uniform Periodic Coefficients.
128. Am, the equations which hitherto have been considered
have had uniform functiona of the variable for the coefficients of
the derivatives ; and the only particular class of uniform functions,
that has been specially adopted with a view to detailed discussion
of the properties of the equation, is constituted by those which
are rational. Many of the properties, however, which have been
established in the preceding chapters, hold for uniform functions
whose form, in the vicinity of a singularity, is similar to that of
a rational function when expressed as a powerseries in such a
vicinity. Among the classes of uniform functions, other than
rational functions, there are two characterised by a set of specific
properties ; viz. simplyperiodic functions, and doublyperiodic
functions ; and accordingly, it seems desirable to consider equa
tions having coefficients of this type. The present chapter will
be devoted to the discussion of equations the coefficients in which
are uniform periodic functions.
Equations with Simplyperiodic Coefficients.
We begin with the case in which the coefficients have only a
single period ; and we take the equation in the form
where p,, ..., p^ are uniform functions of z, are periodic in w,
and have no essential singularity for finite values of 3. Let a
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404; EQUATIONS HAVING [128.
fundamental system of integrals in the domain of any point be
denoted by
/.W. /.W /<4
which therefore are linearly independent. A change of 2 into z+tn
leaves the differential equation unaltered : hence
/.(2 + «), /,(^ + ») /,(^ + «)
are integrals of the equation. That they are linearly independent,
and therefore constitute a fundamental system (it may be in a
new domain), is easily seen ; for
satisfies the equation for aJl values of s, and by making s pass
from any position Z+ a to Z without meeting any singularity, the
integral changes from XcrfriZ + a) to ScrfriZ). If, then, values
of c could be found such that the equation
is satisfied identically (and not merely for special zeros of the
function on the lefthand side), then we should have
2c,/.(^) = 0,
also identically. The latter is impossible, because the integrals
A{z), ■■■,fmi^) constitute a fundamental system; and therefore
the former is impossible. Thus /, (a + ro), . . , , /,„ {e + oi) constitute
a fundamental system.
Suppose now that the domain, in which the original funda
mental system exists, and the domaan, in which the deduced
fundamental system exists, have some region in common that
is not infinitesimal; and consider the integrals within this
common region. As fi{zka), ..., /^(•^ + w) are integrals, and
as/, (e), ...,f,a{z) are a fundamental system, we have equations
of the form
/, (^ + »)  o„,/, w + . . . + «.,/. w )
where the coefficients a are constants ; their determinant is not
zero, because the set of integrals on the lefthand side constitutes
a fundamental system.
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128.] UNIFORM PERIODIC COEFFICIENIS 405
Consider any other integral in this region ; it is of the form
FW = «,/,W+.,/,W + ...+«,A,W,
where /Ci, «2, ..,, «™ are constants; and so
ii' (» + «) = 1 <!„«,/, W + I »„«,/, (2) + . .. + I »„«,/, W,
In order that F(z) may be characterised by the property
^ constant, the coefficients k must be chosen so
l<i^«r = ^«p, (;» = 1, 2, ...,m).
where d i
that
a set of n equations linear and homogeneous in the coefficients k ;
and therefore S must satisfy the equation
(La , a^d, ..., a^
an equation involving the coeSicients a, and so apparently depend
ing ixpon the choice of the fundamental system /i, .,., f^
But, as with the corresponding equation for a set of integrals
near a singularity ( 14), we prove that this equoiion is independent
of the choice of the fundamental system, so that the coefficients
of the powers of 9 are invariants. The proof follows the hues
of Hamburger's proof for the earlier proposition. Let another
fundamental system g^{z), .... ffmi^) existing in the region under
consideration, be such that
?(»■
")in».W + . + f>™9..W. ('■I.'
«}.
the determinant of the coefficients b being different from zero.
The equation, to be satisfied by the multiplier of F(s), is
B(S).
ill  «,
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406 FUNDAMENTAL EQUATION [128.
As the integrals / are a fundamental system in the region, in
which the integrals g exist, we have
5',(3)c,,/,(2) + ...+c,^/™(4 (5 = 1, ,..,m),
where the determinant of the coefficients Cj;, say G, is not zero.
Thus, as
hn9, (^) + . . . + K^g^ = gr{^ + »)
we have
_l lb„c„f, (2) = l_ l^co., /, (,).
This homogeneous linear relation among the linearly independent
integralsy" must be an identity; and therefore
S hraCst — S C^sffst
say. Then
= A(e)C,
BO that, as C is not z
we have
B(0) = Aid),
and the equation is invariantive. We therefore call it the fun
mental equation for the period a.
Let A (z) denote the determinant
4(.). '?::^^ ^^:^' ^":^
1^9, we have
in—
da—' ■ ■
■■ d.'
df,
dz ' ■
df.
■■ di
/.
/. , .
., /
r "fA^)dx
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28.]
>that
FOB THE PERIOD
A {« + «.) I'^'pil')"
where we may assume the integration to take place along a path
that does not approach infinitesimatly near the singularities of pi,
if any. Now, as pi is a uniform function, simplyperiodic in m, it
is known* thatpi is expressible in the form
within such a region as encloses the path of integration ; and the
series is a converging series. But
ly?
if the integer a is distinct from zero ; hence
aw " ■
But, substituting in A(s4w) the expressions for /i (z + w), ...,
/m(^ + f^} ^rid their derivatives, in terms oi fi{z), ..., /™(s) and
their derivatives, we have
A(» + »)
which is the nonvanishing constant term in A (0) ; and thus
In particular, when pi is zero, so that the differential equation
, we have ^„ = ; and then
contains no term in 
A(0) = l^
Kir^™
129. The generic character of the integials depends upon the
nature of the roots of the fundamental equation,
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408 ROOTS OF THE [129.
If the m roots of the fundamental equation are different from
one another, and if they are denoted hy ff^, 0^, ..., ^m, then a
fundamental system of integrals exists, such that
K(^+«)^e,F,(4. (i '»)•
Consider any simple root ffr of the equation A ($) = 0.
Then not all the minors of A {d) of the first order can vanish for
$ = ds', hence m — 1 of the equations
I a,pK,= eKp. (p = h 2, ..., m),
determine ratios of the m quantities k, and consequently determ
ine a function i^r(^) having a multiplier ^,. This holds for each
of the m different roots: and thus m different functions F{z) are
determined.
These m functions are linearly independent of one another. If
there were an equation
7i^i (2) + ^,F,(s) + ...] j^F^ (^) = 0,
which is satisfied identically, thea also
y,F,(s+6y) + y,F,{z+m) + ... + y^F^{z + o,) = Q,
that is,
^ijiF, {z) + d^jj\ (s) + ... + e^'i^F^ {z) = 0.
Similarly,
^1^71 fi («) + ^.'7i^. (s) + . .. + 0^'y^F^ (s) = ;
and so on, up to
d,"^'yiF,(s)+o,^'y,F,{0) + ... + e«,^'r.>^F>n(s) = o.
Now the determinant
W, e,\ 6i, ..., e^'^'l
does not vanish, because the quantities are unequal ; hence
so that the constants 7 all vanish. The m functions F therefore
constitute a fundamental system.
1.30. Next, let °r be a root of ^(^)=0 of multiplicity ^,
where ^ > 1. The equations
^ a,ipKs = 0iCp, (i>= 1, ■■■> m).
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130.] FUNDAMENTAL EQUATION 409
are consistent with one another, though not necessarily inde
pendent of one another : any m — 1 of them are satisfied hy
ratios of the quantities k, which are finite and may contain
arbitrary elements. Giving any particular values to the last, we
have an integral, say 'I>i {z), defined by means of these quantities :
it is such that
and it is a hncar combination of/, (^), ..., f^{z). Taking any one
of the integrals which occur in the expression of this linear com
bination, say /i {z), we modify the fundamental system so as to
replace f^ (s) hy "!>, (a). Let the equations for the increase of the
argument by oi in the modified fundamental system be
/.(^ + t«) = c„*,(^)+c./,(2)4
then the fundamental equation is
fc,,„ /,„(.), (r
'h6,
which, owing to its invariantive character, is A {$) = 0, and therefore
has S for a root of multiplicity /l. Consequently, the equation
has ^ for a root of multiplicity ^ — t ; and therefore the equations
(c^'^)ic^'+c.^ K^' + .. + o^/cJ =0,
Cn^f^!' + 0^3*3' + ■ . ■ + (Cmm ^) K
= 0,
are consistent with one another, and any m — 2 are satisfied by
ratios of the quantities «', which are finite and may contain
arbitrary elements. Giving any particular values to the latter,
and writing
«>. W «■'/. W + «.'/. W + .••+ «..7 W,
we have
■D,(» + «)  x,,*, W + a*. W,
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410 FUNDAMENTAL SYSTEM [130.
where
so that Xsi is a constant, which may be zero. The quantity 0^{e)
is an integral of the differential equation r we use it to replace
some one of the integrals in its expression, say /^is), in the
fundamental system, so that the latter then is constituted by
s.W, i.W/.W /~W
Proceeding similarly from stage to stage, we infer that,
associated with a root ^ of multiplicity /j. of the fundamental
equation, there exists a set of ft integrals such that
0, {^ + w) = V*. (z) + \,,*, (z) + ^*3 (2),
where the coefficients X are constants.
Similarly, if the roots of the equation ^ (^) = are %, ...,&„
of multiplicities ^u.,, .... na respectively, so that /^i + ... + fi,j^ — in,
the fundamental system can be chosen so that it arranges itself in
n sets, each set being associated with one root of the fundamental
equation and having properties of the same nature as the set
associated with the preceding root of multiplicity &.
A function, characterised by the property
is strictly periodic, and sometimes it is said to be periodic of the
first kind. A function, characterised by the property
F(, + „).ilF{z),
where ^ is a constant different from unity, is pseu do periodic, and
sometimes it is said to be periodic of the second kind, 9 being
called its multiplier. A function, characterised by the property
where X and /j. are constants, is also pseudo per iodic, and some
times it is said to be periodic of the third kind.
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130,] OF INTEGBALS 411
With these definitions, the preceding result can be enunciated
as follows*:—
A linear differential equation, the coefficients of which are
simplyperiodic in a period co, possesses integrals which are
periodic of the secmid kind: and the number of such integrals
is at least as great as the number of distinct roots of the funda
mental equation for the period.
Ex. 1. Prove tliat, if the equation
iPw , , .civ! , , , „
integral which is periodic of the third l^ind with a multiplier
e"+^, then
i.i(s+a,)=p,(s)2X,
Hence integrate the equation
shewing that XiB = 4jr^. (Craig.)
Ex. 2. Shew that, if the coefficients in the equation
have the form
p,(.) = 0(.) + ^,
where i^ and yjr are periodic of the first kind, then the equation certainly
possesses one integral that is periodic of the third kind. (Craig,)
131. On the basis of these properties, we can take one step
towards the analytical expression of the integrals.
The integral »I>, (s) is a periodic function of the second kind.
As regards the integral '^^(s), we have
*,(« + '«) ^A^) ■A'
' Floquet, Ann. de Vic. Norm., S^r, 2°, t. sii (1883), p. SS.
that
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412 PERIODIC INTEGRALS [131.
SO that the function on the righthand side is a periodic function
of the first liind, say t/t (z). Therefore
where <I'isa (s) is a constant multiple of *i (g), and the constant
factor may be zero ; and ^^ (z), = i/r (s) *i (z), is a periodic
function of the second kind, with the same multiplier as *i (a).
Vs regards
the integral $3 (3), we
have
*, (s + ffl.
<I>, (2 + 0.
'.if
+w)i
&"*<">■
_x,.x. ,
2SW
. + ^^
2&X.
we hai
80 that ^(3) is periodic of the first kind. Hence
where "t,, (a) = (z) ^^ (a), and therefore is a periodic function of
the second kind with the same multiplier as "^ij where 'i':si(z) is a
linear combination of ^^ (s) and ^, {z), and thus is periodic of the
second kind with the same multiplier as ^i (s) ; and ^si{z) is a
constant multiple of ^, (s), in which the constant factor, viz.
may be zero, and certainly is zero if ^aiz) disappears from (^^{s)
owing to the vanishing of its constant factor.
Proceeding in this way stage by stage, we obtain expressions
for the integrals in succession ; and we find
<^^ {Z) = •3>„ (S) + Z<^ri {Z) + 3= *rs (3) + . . . + 2'^'*rr (z),
where
^ . ( ly X.,riXr.,r s ■  X:.,X^i , .
■^"■W (rlji^'M ''••''
80 that it is a constant multiple of *i{^), the constant factor
being capable of vanishing ; and all the functions 3>^i {z), ^^ (s),
..., *r,ri(2) are periodic functions of the second kind with the
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131.] OF THE SECOND KIND 413
same multiplier as ^^ (z), and are expressible as linear combina
tious of O,, *,i, *3,, ..., <&ri,i. This holds for the values r = 1,
2, ...,(^
Similarly for any other set of integrals, associated with any
multiple root of the fundamental equation of the period.
It may, however, happen that some one of the coefficients
\,a^i vanishes, so that, for all values of r^s, the term in ^„ (a)
disappears. The alternative result is that a linear combination of
the functions ^g{z), ^si{z), ..., "I'lia) can be constructed which is
periodic of the second kind. This linear combination can be used
to replace ^s(^), and thus may be the initial member of another
set of integrals in the group associated with the multiplier &.
The proof of this statement is simple. Assume that \j,j_i vanishes,
and that no one of the coefficients V,.., for yaluesofr^ a vanishes;
and construct the linear combination
choosing the coefficients k so that the term in "J^i disappears and
that the remaining terms are
^ («,*, {z) + «,_,*,_, (2) + . . . + «,*, (z)\.
To satisfy these conditions, we must have
= «,\sl + K^lXs1,1 + .■■ + K4V + fs!^3i + «3^,
= KgXs.ti + K^^'k^l,,^.
Transfer the terms in ics to the lefthand side : the determinant of
the coefficients k on the remaining righthand side is
which by the initial hypothesis does not vanish. Some of the
coefficients Xj,, Xgj, ,.., Xj,j_a are different from zero, for ^i{z)
is not a periodic function of the second kind; hence there are
finite nonzero values for the ratios of «si, ■■, «= to «s When
these values are inserted, let
*.{«)«.*. W + . + «.'i>.Wi
then
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414 SETS OF [13L
so that 'Sfj (s) is periodic of the second kind, with the common
multiplier ^ ; it can replace <E>s (z) in the fundamental sj^tem, and
then can, like Oi(^), be the initial member of another set within
the group of the same type as *i(«), ..., *«i(s). The statement
is thus proved.
132. Any set, such as ^, (s), ..., ^siCs) in the preceding
group of integrals, whether s = /i or be less than /t, can be replaced
by an equivalent set of simpler form.
Let the equation be written
BO that
Also let
P,
SP
P,
3P
and,
generally.
let
P.
d'P
Let the integral of the set containing the highest power of z,
say g''~', be expressed in the form
...4(r l)«0^_, + ^„
the binomial factors being inserted for simplicity. Then, as
F (Z'y}r) = 2P (f ) + KZ''P, (l/r) + «(« l)2=Ps W + ■.■,
we have
= P(».)
^■P(« + ('l)^'P.(« + i('l)Cr2)i>"P,(« + ...
+ <rl)K'P (,(,,)+ (r  2) z'P, (« + ...
+ «r  1) (r  2) {^P (« + ...
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132.] INTEGRALS
which can be aatiefied identically, only if
The first of these conditions shews that
is an integral of the equation. The second shews that
is an integral ; the third that
is an integral. And generally, if w denote
^(Ml}!(r/^)l^
...+(7l)f^^, + 0„
w being an integral of the equation, then each of the quantities
_ _1 a^ 2!(r3)! 5°w (fil)[{rfj.)]3'''w
^"^ riar (r1)! ar^' "'■' (^1)! S?*^"' ■"
is an integral of the equation, when ^ is replaced by £ after differ
entiation. Accordingly, the group of r integrals in the set are
linearly equivalent to
i!^ = .^s + 301,
'ts = 03 4 230a + 2^01 .
M4 = 04 + 3^0, + 32=02 + ^'Vi ,
Ur = ^r + (r~ 1) 30^1 + .. . + (r  1) 3'^02 + 2'<Pu
and any linear combination of these is an integral of the differ
ential equation ; all the quantities which occur in them are
periodic of the second kind, having the same multiplier.
Similarly for any other set ; and thus the vi integrals of the
equation will he constituted hy sets of r^iVi, ..., r„ integrals of the
 m, and the system contains
riodic functions of the second kind.
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416 group of istegrai£ associated [133.
Group of Integrals associated with a Multiple Eoot
OF THE Fundamental Equation of the Period.
133. These results can aiso be obtained by using the proper
ties of the elementary divisors of the quantity A (0), when it is
expressed in its determinantal form. Let the elementary divisors
associated with the root S be
so that, as in § 15, the highest power of ^ — ^i common to all the
first minors of A {$) is (d — 'Sr)"^, the highest power common to all
the second minora of A (6) is {d — ^)'^, and so on; and the minors
of order t (and therefore of degree m — t in the coefficients) of
A (d) are the earliest in successively increasing orders not to
vanish simultaneously when 6 = '^. As in the earlier case dis
cussed in §1 15, 16, we have
Proceeding on lines precisely similar to those followed in  23
for the arrangement, in subgroups, of the group of integrals
aasociated with a multiple root of the fundamental equation
belonging to the singularity, we obtain a corresponding result in
the present case, as follows : —
The group of ft mtegrah aasociated with the root ^ of multipli
city n,belonging to the funda/mental equation for the period at, can be
arranged in r subgroups, where t is the numh&r of elementary
divisors of A {d) which are powers of $~^. If the X members of
any owe of these subgroups be denoted by gi{z), g^iz), ..., gt.{z),
these irdegrals of the differential equation satisfy the characteristic
equations
J. (^ + »)as,w •,
jr. (2 + »)=.*» W+ ft W I
j.(«+")=aftW+».(») r
Taking all these subgroups together, the number of first equations
which occur in them is equal to the number of the sahgroups,
that is, the number of the elementary divisors of A {6) connected
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133.] WITH A MULTIPLE ROOT 417
with ^ — 9 ; the number of second equations which occur is the same
as the number of those indices of the elementary divis&rs connected
with 6 — "^ that are not less than 2 ; the number of third equations
is the same as the number of those indices that are not less tha/n 3 ;
and so on, the number of equations in the first subgroup being
The analogy with the Hamburger subgroups in Chapter ii is
complete.
Corollary. The total number of integrals of the second kind,
defined as satisfying a relation of the form
g {! + «) = eg (.),
where 6 is a constant, is the total number of elementary divisors of
A(0) associated with all the roots of A(d) = 0; a theorem more
exact than Floquet's (§ 130). For the total number of such
integrals, in the group associated with a multiple root of
A ($) = 0, is equal to the number of elementary divisors of A (ff)
associated with that root : and the total number of groups is
equal to the number of distinct roots of A (8) = 0.
134. Some approach to the analytical expressions of the
functions, satisfying the equations characteristic of the subgroup,
can be made, as in § 23. Let
and introduce a difference symbol V, such that*
for any function F; also let
/Xl\ ^ f\l\
G
<^) = x^ + (\') ?;... + (^ 2 ')rxA. + ...
where the functions j(i, ^2, ■■, %>. are periodic functions of 3, with
a period ro, and
V r J rl(\lr)':
* For theae difference symbols in general, i
Mat., See. 2", t. x (1882), pp. 10—45.
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418 GROUP OF INTEGRALS ASSOCIATED [134.
Then if we take
for a!! values of m, we have
holding for all values of n. These are the characteristic equations
of the subgi^oup ; and we therefore can write
with the above notations, for n = 0, 1, . . X — 1
These \ integrals are a linearly independent ''ei out of the
fundamental system; the system will remain fundunental if
ffi! 9i> .^A *re replaced by X other functions Imeailj equivalent
to them and linearly independent of one inother This modifica
tion can he effected in the same way as the corresponding modifi
cation was effected in  24, viz. by introducing a set of functions,
associated with G and defined by the relations
the functions '^^ being periodic functions of z, with tho period w.
Constructing the expressions Vff, V^G, ..., V'^G, we find
V^^G = cx_,,,Gi,
where the constants c are no n vanishing numbers, the exact values
of which are not needed for the present purpose.
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134.] WITH A MULTIPLE ROOT 419
It follows, from the last of the equations, that G^ is a constant
multiple of V*~'{?, and therefore that ^" G, is a constant multiple
of ffi (s) ; we replace y, (s) in tlie fundamental system by ^ G^.
It follows, from the last two of the equations, that G^ is a linear
combination of V—^G and V'~^G, and therefore that S^"(?s is a
linear combination of g^ (z) and gi (s). As g^ {z) has been replaced
in the fundamental system, we now replace gi{z) by ^"Gj; and
the system remains fundamental.
And so on, for the integrals in succession. Proceeding thus,
we obtain X, integrals of the form
?ri9.(2), ^" (?,(«), ...,SiS;^(s).
Further, these integrals are linearly independent, and so they are
linearly eqiiivalent to ^,(3), ^2(2), ...,gx(z). For if any relation,
linear and homogeneous among these quantities, were to exist
with nonvanishing coefficients, we should, on substitution for
Gu G„ ..,, (?;vin terms of G',VG,V=G',.. .,V*G', obtain a relation,
linear and homogeneous among the quantities gi{z), ..., g^i^)
with nonvanishing coefficients. Such a relation does not exist.
Accordingly, the X integrals
can be taken as constituting the required subgroup of integrals.
We now are in a position to enunciate the following result,
defining the group of integrals associated with a multiple root ^
of the fundamental equation of the period : —
When a root ^ 0/ the fundamental equation A (0} = O is of
midtiplidty fi, there is a group of fj, integrals associated with that
root; the group can be arranged in a number of subgroups, their
number being equal to the number of elementary divisors of A {$)
which are powers of "it — 6 ; the number of integrals in the fi/rst
subgroup is equal to the number of those elementary divisors ; the
nwmber in the second subgroup is eqtial to the number of the
exponents of those divisors which are equal to or greater than 2 ;
the number in the third subgroup is equal to the number of the
27—2
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420 GROUP OF INTEGRALS [134.
eicpan&ixts of those divisors which are equal to or greater than 3 ;
and a suhgrmip, which contains X integrals, is equivalent to the X
linearly independent quantities
where
ftWX.+ (''7')x.f+(''2')x.f + 
for r = ], 2, ..., X' the quantities %,, ,.,, y^^ are periodic functions
of z, but they are not necessarily uniform : f denotes — , and
frl\_ (r1) ! _
Note. By taking ;:^„ = q>~"0„, for m = 1, ..., \, and wiiting
the integrals become
e. (^) =4>.+ ('■ ^ ^)>._,^ + r ^ ^) ^^^^ + . , .
the functions i^ having the same character as the functions )(.
135. There is a theorem of the nature of a converse to the
foregoing proposition, which is analogous to Fuchs's theorem
proved in ^ 25 — 28. The theorem, which manifestly is important
as regards the reducibility of a given equation, is as follows :—
If an expression for a quantity u is given in the form
w = ^" \4,n + ^„_i? + 4>„^,K' + ■  + W" + W'l
where & is a constant, all the functions cp^, ..., 0„ are periodic in a>,
and ^ denotes  , then u satisfies a homogeneous linear differential
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135.] CONSTRUCTION OF UNIFORM INTEORALS 421
equation of order n, the coeffidents of which are uniform periodic
functions of z, Iiaviiiff the period a; moreover,
are integrals of the same equation and, taken together with u, they
constitute a f undo/mental system for the equation of order n.
The course of the proof is so similar bo the proof of the corre
sponding theorem as established in §§ 26 — 28 that it need not be
set out here*. It can be divided into three sections ; in the first,
it is proved that — , ..., ■ ^ satisfy such an equation, if u
satisfies it ; in the second, it is proved that these must form a
fiindamental system, for no homogeneous linear relation with non
evanescent coefficients can exist among them ; in the third, it is
shewn that the linear equation, which has these quantities for its
fundamental system, has uniform periodic functions of z with
period w for its coefRcients. The details of the proof are left to
the student.
Mode of obtaining Integrals that are Uniform.
136. The further determination of the analytical expressions
of the integrals, on the basis of the properties already established,
is not possible in the general case. Thus the functions )(i, ..., %a>
occurring in the subgroup specially considered in § 134, are
periodic functions of the second kind with a multiplier ^. If we
take new functions ■^i{2), ..., "^/.(f), such that
these new functions are periodic of the first kind. But further
properties of the functions must be given if there is to be any
further determination of their form.
When we limit ourselves to the consideration of those equations
whose integrals are uniform functions, (criteria are determined
* Some of the analysis of g 133 ig useful in establishinf; the theorem.
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422 EQUATIONS HAVING [136.
independently by considering the integrals in the vicinity of the
singularities), some further progress can be made ; bub, of course,
the assumption that the integrals are of this character must be
justified by appro)riate limitations upon the forms of the coeffi
cients Pi, ..., pm in the original dift'erenbial equation. In such
cases, every quantity such as ^^C^) i^ a uniform simplyperiodie
function of the first kind; it can therefore* be expressed in the
form of a Laurent Fourier series such as
Such a, form of expression does not lead, however, towards the
determination of the criteria for securing such a result or any
other result of a corresponding kind for any other assumption. In
particular examples, we adopt a different method of practical
procedure.
In order to determine some of the functional properties of the
integrals, it frequently is expedient to change the variable so that,
if possible, the transformed equation belongs to one or other of
the classes of equations considered in preceding chapters.
Thus if the coefficients^!, ...,pm, which are uniform periodic
then, introducing a new variable (, where
we obtain a linear equation, the coefficients of which are rational
functions of t. Some characteristic properties of the integrals
of the equation in the latter form can be obtained by earlier
processes ; it may even be possible to determine the fundamental
system of integrals.
The preceding transformation is, however, not the only one
that can be used with advantage ; and it often happens that
the special form of a particular equation suggests a special
transformation which is effective. In pai'ticular, if the coeffi
cients in the equation are alternately odd and even functions,
' T. F,, % 112,
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SJMPLYPERIODIC COEFFICIENTS
such that pi,pi,Pi, .. are odd, and p2,pi,pe, ■■■ are even, then
wc may take
as a new independent variable r it is easy to prove that the
transformed equation has uniform functions of ( for its coeffi
cients. Also, some indication is occasionally given as to a choice
between these two transformations ; for example, if an irreducible
pole of the original equation is a = 0, we should choose
( = sin —
aa the transformation, and consider the integrals in the vicinity of
t = 0; whereas the other would be chosen, if an irreducible pole of
the equation is s = ^oi.
Another transformation, that sometimes can prove effective, is
any uniform function of 3, periodic in o>, can be expressed as a
uniform function of t ; and the differential equation is transformed
into one which has uniform functions of t for its coefficients.
Ex. 1. Couaider the equation
J'+2.:^cot.+ (6 + .oot..)„.0,
where a, b, c are constants. Writing
we have the equation
whei'e
and the oquatio
+ (/3+Tcot2s)», = O,
The indicial equation for t—0 is
p(pl) + y=0.
If v=Saj''*''
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424 EXAMPLES [136.
satisfies the equation, we have
n2+2n(p2) + 403p
'• if+M^ir^) *■"'■
The form of a„, in terms of o^a, shews that the series for y converges for
values of ! ^ 1. If the two roots of the indicia! equation arc p^ and p^, and
f{i,Pi),f{t, Pa) ho the two values of y, the primitive of the original equation is
,>— in. J/(siii 4 p,)+il/(«n ,, a)}.
Ex. 2. Consider the equation
we find the transformed equation for i to be
which is Legendre's equation and so its primitive is
Ex. 3. Obtain int^rals of the equations
j5 +J cot!wcoseo^3=0;
d^ dz
d^ die a a a —ft.
dz^ da ~ '
(i)
(ii);
()S(snr^OSHovk.i.^)^
Ex. 4. One integral, /(^), of the equation
4(i!,i„.)5 + !.(3.m. + 2c„..^«)* + (53co..,m.),.0
the relation
find the general solution. (Math. Tripos, Part it, 1896.)
Ex. S. Shew that the equation
has an integral
where S[ has an appropriate constant value; and obtain the primitive.
(M. Elliott.)
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136.] liapounoff's investigation
Ex. 6. Obtain an integral of tlie equation
where A is a constant, in the forin
^_sin(?gi) sin(a33) ^(cots^+cots,)
where 2, and e^ are appropriate determinate constants ; and obtain the
primitive. (M. Elliott)
E.V. 7. Integrate the equation
where w is an mt«ger, and A is a constant. (M. Elliott.)
137. A somewhat different form of the theory is developed
by Liapounoff*, whose investigation deals with a more general
equation, given by
where /i is a parameter, and p (s) is a uniform periodic function of
period €0.
Let f(z) and ^ (e) he two integrals of the differential equation,
respectively determined by the initial conditions
/(0).1 *(0).01
/'(o)=or f(o)=ir
Then we have relations of the form
and the equation for determining the multipliers is
(nn)(n8)^7 = 0,
that is,
a=  (a + 5) fi + 1 = 0,
as in § 127, Ex. 1. Clearly, we have
* Comptes Hendus, t. cxxm (lS9e), pp. 1248—1252; ih., t. cxSTiii (1899),
pp. 910—913, 10851083.
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426 liapounoff's [137.
BO that, if we write
the equation is
Writing
;"24n + 1.0.
= A+(A'l)t,
and assuming that A'—l does not vanish, we obtain two integrals
in the form
where Fj (s), F^ {z) are functions of z, periodic in to ; and thus the
complete priniitive of the equation can be obtained. The actual
expressions for J'i(s) and F^{z) can fee constructed as in the
preceding sections ; and the value of p depends upon that of A .
When ^ = 0, the primitive of the original equation is
shewing that the equation for dotermining the multipliers is
(li 1)^ = 0;
and then A = \. Hence, when /a is not zero, and when A is
expanded in powers of /*, it is inferred that A is oi the form
^ = 1 — fi,Ai + f>?A^ — iJ?A,^ + ....
When .^1, A3, ^3, ... are known, the two values of il, which
satisfy the equation
n'^24 11 + 1=0,
can be regarded as known, and the primitive of the dift'erential
equation can be obtained.
For the purpose of obtaining the value of A, which is
4 = i(/W + f(»)l.
where the integrals f{z) and <^ {s) are deiined by the initial
conditions, we assume both f{z) and {z) expanded in powers of
f{z) = Wo + fiMi + ij?u.i + . . . ;
then, in order that it may satisfy the equation
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137.]
EQUATIONS
we have
and so on
, From the first, we have
Ik
= a,
, + *.«;
from the
second,
we
have
from the third, we have
= ai + b,z — j rfy I u„(a:)p{x)da:;
ve have
= C!a + 632  1 dT/j «i (x) p («) da: ;
and so on. Now
/(0)=1, /'(0) = 0;
accordingly,
a^ + /j.a, + f>?a.i + . . . = 1,
ha + iih^ + f.% + ... =0.
Taking account of the fact that /j, is parametric, we have
«„ = !, a, = for s = l, "2, ..., ?i, = for s0, 1,2, ,
and thus we have
M„=l,
Wi = j (^^f p(a:)da:,
and so on. The value oif(z) is given by
/(s) = 1 + ^Mi + /t'Ms + . . , .
Similarly for (2), which is determined by the conditions
<^(0) = 0, .^'{0) = 1;
its value is given by
<j>{s) = e + fiVi + /iH'a + . . . ,
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428 liapounoff's [IS?,
where
Vi = j dyj xp{a;)dx,
V, = \yy\yia:)v,{x)dx.
and so on.
We require the quantities /(w) and '^'(oi): let thorn be
denoted by
where
[/j = — I dy\ p{x) dx,
T, = — I cnp{(£) dx,
and so for the others. Substituting the value of ^ in the form
4 = 1 /tjl, + /iMi. — ...,
we have
= 1 dy\ p{x)dx+ i xp(x)dx
= j %j ^(«)'^ic + j yp(y)dy.
But
^ 1^ f^^i? (a') (^a^l = f V (*) ^^ + yp iy) ;
integrating between the limits and a, we have
Next, we have
iA, = 11.r,
To transform these definite integrals, we write
j'"j,W<faPW, P(»)=n.
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137.] METHOD 429
so that
uAi!) = t'dtf'p(6)de = r P{t}dt,
v,(a:) = !'dtj'Sii{e)de
~rtP{t)dt+rdtl'p{S)de
= {'dti' {F{lf)P(t)]ie.
We have
J 1"! {s)j^p(s) d!/\  «. (y)p to) P'(s);
therefore
and thus the first integial in tlie expression for 2A^ is equal to
j'dyj'{F{g}F(a:)]P(^)d^.
Similarly, we have
therefore
= nj'\P(!,)% + n^"<!yJ'p(a.)(i»
+ j''yP(y)*JJ<ij/'P&)PW<i«:.
The first and third terms on the righthand side together are
JJ(nP(y)P(y)t,<iy
= /J.iy/VP(y)lP(y)<fc.
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430 liapoukoff's equations [137.
ao that
I" V, (:v)p (.^) <^ =  /^" d^j' [ii'P (y)} \P iy)  P (.^)} dx,
which gives a traasformation of the second integral in ^A^.
Combining the results of transformation for the two integrals in
2^2 , we have
HA, = 1^°' d>j r {ilP{y) + P {x)} {P (y)  P (^)i dx.
Similarly, it may be proved that the value of ^A^ is
j'Ji/J%/^(nPW + PW!PWP(y)HP(s)PWi<fc,
and so on i so that the value of A, and therefore the value of P, is
known.
The investigation is continued by Liapounoff, especially for
the purpose of discussing the values of fi. which satisfy the
equation
^=1 = 0;
aad the results appear to be of importance in the discussion of
the stability of motion. The reader is referred to the notes by
Liapounoff already cited (p. 425, note) ; other references to more
detailed investigations are there given.
Esc. 1. Establish by induction, or otherwise, the general law for the
coefficients A, viz.
2.t„= <h!,\ rf%... I @rf«„,
where
©.(iiP(i,)+P(«.,)){fWPW]fi'WP(».))... If ('.Ji" (».)>■
Ex. 2. Shew that, if the peiiodic function p(s;) always is positive, then
all the coefficients A are positive; and prove that
Hence shew that, when p (x) is positive and satisfies the inequality
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137.] EQUATION OF THE ELLIPTIC CYLINDER 431
Ex. 3. Prove that, if the periodic function p {m) be real and odd, so that
the series for A contains only even powers of /t, then
A^= 4 I °'(fej r'dx^ fds;^ i"' [P^ P^f {P.^~ P^f dx^,
and MO on, where P,. denotes i'i^r) Prove also that, if
tho constant a being determined so that I PciK=0, and if
"" P''dx:^i,
'I?
then ji^<I.
Ex. 4. Discnss the values of )i which are roots of the equation
(All these results are due to Liapounoff.)
Discussion of the Equation of the Elliptic Gylindbe.
138. One of the most important equations of the ciass, which
has been considered in § 137, is the equation
commonly called the equation of the elliptic cylinder; it is of
frequent occurrence in mathematical physics and astronomical
dynamics. It forms the subject of many investigations* It is
known (g 55) to be a transformation of the limiting form of an
equation of Fuchsian type. Moreover, it has already ( 127, Ex. 1)
been partially discussed in connection with another equation and
for another purpose. In this place, it will be brought into relation
with the preceding general theory.
Let new independent variables u ov v h& introduced, such
that
* Heine, Handbuch der Kugelfanctianen, 1. 1, pp. 404 — 415; Lindeniann, Math.
J)iji.,t. XXI! (1883), pp. 117—133; Tisaerand, Mecanique Cilcste,t. in, oh. i, at the
end of which other referenoes are given.
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432 EQUATION OF THE [138.
The equation becomes
.d^w , ,, „ , dw , , ^ ,
"* *^ ~ "^ rf^? "^ *^^ ^ ^""^ rf^ + ^^"^ ~ ' + ^'^''^ "^ = "^'
when u is the independent variable ; and it becomes
, dhi) , ,, „ , dw , , ^ ,
when V is the independent variable. Accordingly, if
is an integral of the equation, another integral is provided by
»=/(., c).
The indicial equation for m = is
p{/'^) = 0;
if
w = ta^vP+i'
be the integral, the scale of relation between the coefficients ap is
(p + p)(p + p^)S=KP + pl)°i(ac))api^cap_a,
with the relations
«o= 1,
When p = 0, let fflp = ^ {p, c) ; when p = ^, let aj, = ?y (^, c). Then
two integrals of the equation are
X, i u^e{p, c), a^i= i ?!P+sa(^, c),
M'ith the convention
t)(0, c) = l=Sy(0, c).
It is clear that, when z is Jtt, so that m is 0,
..=,, J=o,
' ' dz
1.
r, as the equation in w i
s satisfied by »
„ and
dx, dx
C
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138.] ELLIFHC CYLINDER 433
But (^M =  2 Bin s COS ads =  2 [u (1  u)\idz, so that
' ds " dz
When 2 = ^7r, the lefthand side is equal to 1 : hence
and therefore, for all values of z, we have
dxa dxi __
^ dz " dz
Two other integrals of the equation are given by
they are such that, when z is 0, and therefore v is 0,
■^° dz ^ dz
and, for all values of s,
dyi dtig
■^^ dz ■^ ds
Now when z is real, both m and v are real and lie between
and 1 ; and, in particular, when 3 = ^7r, then « = •«= . For such
values, x„, x,, y^, y,, coexist ; and so we have relations of the form
where a, (S, /, S aie constants. Hence
y. (i) = ™. G) + /3a>, (i), </.'(« = «; (i) + Hi'l (J),
where
and so for the others. Hence
« = j.(i)<a)y.'(i)«.(i).
/3= y.^^/ffi + y.' (»■».&)■
Similarly
7 = </.(!)<(» y.'(i)''.(i)l
S= J(.tt)»=.'(i)+!/.'(l)«.(J)l'
F. IV.
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434 EQUATION Of THE [138.
and it is easy to verify that
Moreover, we have
■*» = %!.  /3i/il
^1 =  7^0 + «yJ ■
139. The integrals x^ and a;, are valid in the domain of w = ;
tho integrals y„ and y^ are valid in the domain of v = 0, that is, of
u= 1. Lindemann* proceeds, as follows, to obtain uniform inte
grals valid over the whole of the finite part of the plane.
After a small closed circuit of u round its origin, x^ returns
to its initial value and «, changes its sign ; hence y^ becomes
aic„— /3«], and j/i becomes yxn — hxt. After a small closed circuit
of u round the point 1, the integral i/o returns to its initial value
and y, changes its sign. Consider a quantity ij, where
jj = Ay^ + %l^
as a function of u. It remains unchanged when u moves round
the point 1. Its two values in the vicinity of it = are
{Ax' + £7=) x^ + {A^ + £8=) x^ + 2 {Ai0 + B7S) x,x^,
{Aa^ + Brf) x," + {A^^ + BS') x/  2 (Aa^ + SyS) x,x„
which are the same if
Aa^ + By& = l}:
henee the function is uniform in the vicinity of u = if this
condition is satisfied, that is, the function is uniform over the
whole plane.
The condition is satisfied if we take
and then
n = tt^y,"  ySyo".
Moreover, in the region of existence common to y^, y,, x„. a;,, we
'^^yi' — 7^3/11° = 0Sxi^ — ayxa\
Hence defining the function 71 in the domain of 3= by its value
in terms of y^ and j/i , and defining it in the domain of 2 = 1 by its
value in terms oi x^ and a'i, we have a function
^ = F{u) = F(>Ms's)=^ (i),
• Math. Ann., t. xxii (1883), pp. 117—123.
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139.] ELLIPTIC CYLINDER 435
say, which is regular in the vicinity of m = 0, regular in the vicinity
of w = 1, and therefore is regular over the whole iinite part of the
zplane. Now let
F,y,(a/3)t + s.(7S)ll
7.!,,(.«)lS.(,S)tf'
then
= 2(=i/37S)l.
Also
and therefore
l'«t+5^.?^" = <E>'(^)
ds dz
Hence
Y, dz ^ ^{z) <E>(2) '
1 dY,_,^'(z) (a^78)'.
and therefore
where
These integrals of the original differential equation are valid over
the whole of the finite part of the plane. Accordingly, we may
take two integrals
M {
(?,(2) = {a>(2)j4e' '*w
as integrals, which are valid over the plane and have z—'X for
their sole essential singularity. We now proceed to shew that
they are uniform over the plane.
Substituting in the original differential equation, we have
{a + c cos Is) <^'  1^'' + ^4><l>" + M=  ;
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436 lindemann's [139.
so that, as if in general is not zero, any root of 'I' = is a simple
root. Let k denote such a root : then
Now let z describe a simple closed contour, including k and no
other root of $ = 0, and passing through no root of "5 = 0. Then,
at the end of the contour, *^(2))S has changed its sign. As for
the exponential factors in G{z) and 0,(z), they are multiplied by
respectively, the integral being taken round the contour, that
is, they are multiplied by
^m!'}^,
{k)
that is, by —1. Thus G{z) and G^(s) are unaffected by the
contour; they are therefore uniform in the vicinity. Moreover,
in the immediate vicinity of k, we have
^(0) = (zk}^'{k) + ...,
so that
GA^)={<^'(k)]^(^k)e^^'''^Q(zk),
so that A: is a simple root for one of the integrals and it is not a
root for the other. Similarly, in the vicinity of any other root of
^ = ; hence G and G, are uniform over the whole plane.
Now take any path from z to s + tt, for tt is the period for
the original equation. We have
where F is uniform ; hence
i,(,+ ^)—H^), {*{^ + ^))4 = (l)'jO(2)jS,
where r is or 1, depending upon the path from to tt. The
effect upon the exponential factor of G (e) is to multiply it by
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139.] METHOD 437
Wo know that ^{s) is regular over the whole plane, that it is
periodic in ir, and that it has only simple roots ; hence, taking a
path between z and a + tt, that nowheie is near a root, we can
1
valid everywhere in the range of integration. Then
and, consequently, if
then
Similarly
Hence G and G^ are the two periodic functions of the second kind,
which are integrals of the original equation* ; and they have been
proved to be uniform functions, regular everywhere in the iinite
part of the plane.
Es. Shew that the equation
has two particular mtegrals the product of which is a, singlevahied tnius
cendental function. F{z) ; acd shew that the integraLs are
"''"•""'■"■["/ (.(i^'i^w ]
where C is a determinate constant. In what circumatatices are these two
particular integrals coincident ^ (Math. Tripos, Part ii, 1898.)
liO. The multipliers /m and — are thus the roots of the
equation
D.^ in + 1=0,
' This ineluaion of Lindemann'a apeeial result within the general theory is dua
to Stieltjfls, AetT. Naehr., t cii (1884), pp. 147, 148.
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438 INVARIANT OP THE EQUATION [140.
where the invariant / of the period o> is
Another expression for this invariant, consequently leading to
another mode of obtaining these multipliers, has already been
given in Ex. 1, § 127. Both processes are dependent upon the
determination of simple special solutions of the original differential
equation.
Another method of proceeding is as follows. Let
so that
so that, if
(?(£)=e*''©(3),
then, as 6(z) is a uniform function of a, regular over the whole
plane, (s) is a uniform periodic function of the first kind, regular
over the whole plane ; and ir is the period. Hence we have
and therefore
Now in the vicinity of a = 0, the integral y^ is even and yi is odd ;
hence G(z) contains both odd and even parts. The form of the
differential equation shews that, if /(z) is an integral, then f{—z}
also is an integral ; hence, as (a) exists over the finite part of
the plane, G(—z) also is an integral. Henco, taking
where a is an arbitrary constant, it follows that Il{z) is an integral
of the original equation, which exists for all finite values of z.
Substituting in the differential equation, and noting that
cos 2s cos [(2m + h)z + a\
= ^ cos {(2n 2 + h)s + ix} + ^cos [(2« + 2 + h)s + a.\,
we have
"s" i{a  (2n + hy\ K„ + ^v (>c„^, + Kn+,)'] cos {(2n + h)s + u}^0.
yGoosle
140.] OF THE ELLIPTIC CYLINDEK 439
as an equation which must be identically satisfied ; hence
{a  (2k + hf] «„ + Jc (k„_, + K„+0 = 0,
for all values of n from co to + a^ .
The mode of dealing with this infinite set of equations by
means of infinite determinants has been indicated in a preceding
chapter, and much of the analysis of the first example in § 126
is directly applicable here : so we shall not further discuss this
mode of obtaining h and the ratios of the coefficients «. There is,
however, another method of obtaining these quantities: it is due
to Lindstedt* and is specially adapted to the differential equation
under consideration, for purposes of approximation when c is con
veniently small. Writing
«,. = 2 (2ii + hy  2«,
?'
C
1
c
<Xn
a„a„+, a„+,a„+<j
1
1 I
Owing to the form of — for increasing values of r, it is easy to
prove that this infinite continued fraction converges, for all values
of n. We therefore have
Similarly
K„ 1  1  1 
inf
c t= c'^
«_„ o_„a_„_i a_„_ia_n.
'■ ... ad inf..
,, 1 1 1
1. dn I'Acad. Si Pitersbourg, t. xxx
1 (18B3), No. 4.
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440 lindstedt's [140,
which is a converging continued fraction ; and, in particular,
c c" d"
/Co 1 — 1 — 1  ■ ■ ■
But, from the fundamental difference equation,
therefore
g2 g3 (.2 p2 p! C^
1 _ °^»"i °i^ C2W3 , °ut(i 0.^1 a_j a;0:;
1  1 l"""^ 1 1 1 ■■■'
a transcendental equation to determine h, which of course is
equivalent to the corresponding equation arising out of the
vanishing of the infinite determinant 1) (p).
Denoting the first continued fraction by  and the second by
^ , so that these values may be regarded as coiivergents of infinite
order, we easily find
r=a «=r+a (S+2 OrOr+i a,a^i Ojat+i J
o = 1  i ^ + i i ^ ''' ■
si i — ' ^^L.+ ...;
r=lj=r+2 (=8+2 ar«r+I aa«s+i OfOi+i
the values of ^' and q' are derivable from the expressions in p
and q respectively, by changing a^ into «_„ (for all values of /t)
wherever a^ occurs.
The equation manifestly lends itself easily to successive ap
proximations. Thus, if we neglect C and higher powers, we have
which, to this order of approximation, gives
The calculation of the coefficients can similarly be effected.
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140.]
METHOD
Piove that, up to sixth powers of c inclusive,
< i , 1 <^
1024c[2(la)3(4a)
105a'H55a'+3815w=4705aHl653(t
(In astronomical applications, a it
pared with a.)
Ex. 2. Taking k^=\^ and writing
prove that, up to e^ iuclusive,
1(1.)' (4
mally not ai
ger, and c is small com
(Poincar^, Tiaserand.)
U+? 10243(l+j)9(2 + s')(lg)r
^\\q 1024j(I9)'(2y)(l+2Jj
(c)=f coa(Z+fe) , _oo8(£4^)_l
■^ 2! l(l + j)(2+j)'^(ly)(23)J
(i«:^f c oa(g+6. ) , eoa{Z_6^
3! \(l + 5)(2+3)(3 + j)
whore y^^
"(lj)(2S)(3y)J'
(Poincar^, Tisserand.)
Ex. 3. In the investigation of § 138, the quantity if is supposed to be
different from zero. When M is zero, the integrals Q{z) and Q^if) aro
effectively the same ; and neither of them is uniform, so that the remainder
of the investigation does not apply.
Discuss the case when j¥=0. (Heine.)
Equations i
[ Uniform Doubly periodic Coefficients,
141. We proceed now to the consideration of linear equations,
the coefficients in which are uniform doublyperiodic functions of
the independent variable. Let the equation be
= 0,
where p^, ..., p,„ are uniform functions of z, which have no
essential singularity in the finite part of the plane and are
doubly periodic in periods to and w', such that the ratio of oi' to w
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442 DOUBLYPERIODIC [141.
is not purely real. A fundamental system of integrals exists in
the domain of any finite value of z, and may be denoted by
/.W. /.(') /«,
which accordingly are linearly independent of one another.
The differential equation is unaltered when 3 + lo is wiitteii
for 2 : hence
/,(^+«), /,(^+„) /„<,+»)
are integrals of the equation and, as in § 128, they constitute
a fundamental system of integrals. Similarly, as the differential
equation is unaltered when a + to' is written for 3,
/.(«+»'), /.(^+»') /■(^+»')
constitute a fundamental system of integrals. Choosing therefore
a region common to the domains of these three fundamental
systems (a choice that always can be made because the singu
larities of the integrals are isolated points, finite in number
within any limited portion of the plane), we have relations of the
form
/, (« + «) = o„/,(^) + ...+o„/„W
/. (^ + " )  «»l/l («) + ...+ 0,.»/,(2)
and
/. (^ + .)_4,. /.(.)+. .. + i,./„W
/,. (2 + »■) . J,,./, W + . . . + (.„/. W )
valid within the region chosen. The coefficients a are constant,
and their determinant is not zero ; the coefficients b also are
constant, and their determinant also is not zero. The two sets
of relations may be represented in the form
/(^ + »)s/w, /(^ + »)=.S7W,
where S and S' denote the linesir substitutions in the relations.
The coefficients in the two substitutions are not entirely
independent of one another. We manifestly have
/,((^ + .) + »l/,l(^ + »■) + !,
for all values of r. The symbolic expression of this property is
/((. + ») + ..■) = S'f(z + „ ) = S'iv w,
/!(. + «■) + « I = s/(« + »')ss7W,
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141.] COEFFICIENTS 443
80 that
or the linear subsfcitutions are interchangeable. The explicit
expression of relations between the constants is obtainable from
the equation
br:A (^ + «) + &,./. (^ + a>) + . . . + 1™/,„ (^ + o>)
= a.,/, (z + 0.') +«„/, (^ + »')+... + a™/« (^ + o>'),
by substituting for the functions / (3 + w) in the lefthand side
and the functions /(z + o>') in the righthand side. The result
must be an identity, for otherwise there would be a linear relation
between the members of the fundamental system f,(2;), ...,/„(«);
hence, comparing the coefficients of fg (z) on the two sides after
substitution, we have
aay. This holds for the m" equations that arise from the values
r, s= 1, ...,m. Of the m' equations, only m^ — m are independent
of one another, a statement the verification of which (alike in
genera!, and for the special values m = 2, m = 3) is left to the
reader : it can also he inferred from some equations which will
be obtained immediately. The number of the relations is less
important, than their existence and their form, for the establish
ment of Picard's theorem relating to integrals with the character
istic property of doublyperiodic functions of the second kind.
Consider a linear combination of the members of the funda
mental system in the form
F(^)\A(z) + KMz) + ...+K,.f^{^),
where X, will be taken as equal to unity when it is not bound to
be zero; and let the constants X^, ..., X^ be chosen so that, if
possible, the relation
is satisfied, 6 being some constant. To this end, we must have
\,0 = \a^^ lXjaai +X3asi + ... 4 X,„a,„ ,
X^0^
n + X3%,„ 
fX,„am„„
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141
and therefore
FUNDAMENTAL EQUATIONS
[141.
the equation satisfied by 0.
As in the case of the single period in § 128, it may be proved
that this equation is independent of the original choice of the
fundamental system of integrals /i(z), ..., fmi^) The coefficients
of the various powers of 6 are therefore invariants, and the equa
tion is called the fundamental equation for the period o).
Now let
= \^1W
\A^ + \shsm
I \»b,„rn
Multiply the earlier equations, which define the quantities X and
lead to the equation fL(B) = 0, by b^r, b^, ■■■, &«»■ respectively, and
add: then
Ofir = \ (Oii fcir + «12 tsr + ■ ■ ■ + tim f>mr)
+ 'K (<*!! 6,T + «32 6sr +  ■ ■ + Ossn hmr)
+ Ki (Clnil &ir + ChaAr + ■■■ + amm^mr)
= \i (b„ a,^ + bi3 a^ + .+hm Omf)
+ Ki (6r«l«]r + h^a^y + . . . + imma,nr)
This holds for all values of r ; and thus we have
nr" = '^ifim + ^ «»ni + ^ ttsni + •■■ + "3'" '^mitf
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14.1.]
When the
uniquely
FOR THE PERIODS 445
; compared with the earlier equations, we have
for all values of r; and therefore the same values of X^, ..,, X„,
that enable the equations connected with the period fu to be
:ad to the equations
X„^=Xi6im + X2ijm + ?^3&3m+.+X^&mm.
Hence
J?(2 + «)X,/,(« + »') + X.,/.(^+«')+ + V/,.(^ + «')
 9' (A,/, w + x,/, (^) + . . . + \,u Wl
on using the prece
satisfies the equati
Q.'{e') =
ing equations. Moreover, th
n
b,^e\ b^ , ..., b,„,
b,i , h^ff, ..., i,„a
IS multiplier 6
= 0.
h«. , b^ i™„^'
This equation, like il {B) — 0, is independent of the initial choice
of the fundamental system of integrals /i (z), .,., /m{^), the proof
teing similar to that in  128. The coefficients of the various
pow&rs of 6' are therefore invariants; and the equation is called
the fmidamental equation for the period «'.
The term independent of ^ in II {$), and the term independent
of 6' in il'{ff), can be obtained simply. Let A (a) denote the
determinant
d'A ■< "/■ d'^A
da™' ' ds™^ dn^'
AK
df,
dz
f.
dU
dn
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446 FUNDAMENTAL EQUATIONS [141.
then, as in § 9, we have
.. —  .
Hence
I p, Ix) dx
so that
A (z + ») _ f'*"r, (•)<!•
aw "~" '
and similarly
where we manifestly may assume that the path of integration
does not approach infinitesimally near the singularities of jj,.
Now 'pi is a uniform doubly periodic function with no essentia!
singularity in the finite part of the plane; if, therefore, a^, ,.,, a„
denote its irreducible poles, and if f (s) denote the usual Weier
strassian function in the same periods w and a' as Pi, we have*
with the condition
Now
j'*'p, W d^=C^+SA, log ^(t " v""
+ l^ii,{ir(2 + „«,)f(^«,)l
Co.+ I ^,i7r + , ,0 + 2, («»,)!+ I 2,i(,
= Co)  2, S 4,0, + 2, i a,  D,
' r. 7''., g 129.
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141,] OF THE PERIODS 447
say ; and similarly
say. But, substituting in A (s + w) the expressions for f^ (z + a),
..., /m (2 + w) and their derivatives in terms of y, (jr), . , . , f^^ (z)
and their derivatives, we have
which is the nonvanishing constant term in il (0). Thus
and similarly
il'((9') = e»' +
Kir^'™
In particular, when p^ is zero, so that the differential equation
has no term in ^r~ — r , we have I> = 0, D' — Q; and thon
il(6) = \ + ...+ ( 1)"^, il'(^')= 1 + ...+( 1)"!?"".
Integrals which are Doitblyperiodic Functions of the
Second Kind.
142. Let ^ be a root of the equation SI {&) = 0. Then quanti
ties X,, ...,Xm exist such that the equations leading to ii(^) =
are satisfied; and a quantity 6' is obtained, when the values of
Xu, ..., X,n are substituted in its expression. It thus follows
that there is an integral F{z) of the differential equation such
that
F{z + a,)^6F{z), F{z + o>') = e'F(z),
where $ and d' are constants. Such a function is called* doubly
periodic of the second kind : and therefore it follows that a linear
differential equation, which has uniform doublypetiodio functions
for its coefficients, possesses an integral which is a doubly periodic
function of the second kind: a result first given by Picard.
* T. F., g lae.
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448 picard's [142.
When d is si simple root of the equation fi (S) = 0, then
Xu, ..,X™ are uniquely determinate: and 6' is uniquely determ
inate. When S is a multiple root of its equation, quantities
"Kt, ..., Xm exist satisfying the associated equations but they
are not uniquely determinate : and assigned values of X^, ., X™
determine ff. Similarly for ^ as a root of the equation li' (6") — 0.
Combining these results, we have the theorem* i
A linear differential equation, having doublyperiodic functions
for its coefficients, possesses at least as many integrals which are
doublyperiodic functions of the second kind as either of the equa
tions li (0) = 0, H' (6') = has distinct roots.
By using the elementaiy divisors of fl {$) = 0, we can obtain
a more exact estimate of the number of integrals which are
periodic functions of the second kind, associated with a multiple
root.
Let 6, be a root of il{6) = of multiplicity Xi, and let ii^ be
the number of different elementary divisors of H (6) which are
powers of ^ — ^j, so that the minors of il (0) of order m, are the
first in successively increasing order which do not vanish simul
taneously when = $,. Then (§ 133) the number of integrals,
which satisfy the equation
IS
precisely equal to «i.
* These equfttions appear to have been oonsidered first by Pioard in eenoral;
see Comptes Remlris, t. sc (1880), pp. 12&131, 2<)3— 295; Creile, t. xc (1880),
pp. 281—303.
Their properties were farther developed by Floquet, Gomptes Bendvs, t. xcviii
(1634), pp. 82—85, Arm. de V^b. Norm. Sup., 3™ Sir., t. I (1884), pp. 181—238,
\Thich should be consulted iu oouueotioii with many of the following investigations.
A proof of Picard's tbeorem, different from that in the test, is given by Barnes,
Messenger of Malltematici, t. sxvii (1897), pp. 16, 17.
Investigations of a difierent kind, leading to equations the primitives of which
are espreasible in terms of doublyperiodic functions, are curried out in Halphen'a
memoir "Sur la reduction des equations <liff4rentiel!es lin^aires aux formes
int^grables," Mim. des Sav. Strang., t. xsvni (1883), No. 1, 301 pp. ; particularly,
chapters ii and ii.
The most important equation of the type under consideration is the general
form of Lainfi'a equation. It had heen considered by Hermite, previous to Picard's
investigations; and it has formed the subject of many inemoirH, references to some
of which will be found in my Theory of Fumtiois, ^% 1S7— 141.
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14;2.] THEOHEW 449
Moreover, in that case, tii of the equations in § 141 for
determining the quantities X are dependent upon the remaining
m — Ui. Let the last m — Ui be a set of independent equations,
determining Xn^+,, ..., \m in terms of \, Xg, ..., X„^ ; and suppose
that the expressions are
Xj = /;,iXi + k^\! + kgs\+ ... + km,'K„
for s = «, + !, n, + % ..., m. Then
F (3) = \A (2) + X./. (2) + , . . + X„./,„ (s)
= \(fi is) + ^^9^ (^) + . . . + X„, 3,,, {2),
where
<,,(»)=/,w+_J^^t/.(.),
for r = l, 2, ..., n,; and each of the functions g^, ...,^,i, is snch
that
j,(2+«.)e).s,(.).
Ah regards the possible multiplier 6^ for the other period, we
have
6^ = Xii„ + X2&„i + X,6s, + . . . + X,„J™i
= x,A + M, + ...+x„,s„,,
say, where
and the effect upon F {s) of the increase of argument by the
period <o' is given by
Now 6i is not zero, for it is a root of O' {&) = which has no
zero root ; and therefore not all the quantities B^, B^, ..., Bn, can
vanish. Let Bi, B^, ..., B, be those which do not vanish ; then we
have
X.sf, (s + 0.) + X^g^ (^ + w) + . . . + X„,^„, {z + <o)
= (XiS, H X,B, + ... + X^B,) [\g^ {e) + \^g^ (s) + ... + X„,^„, (s)}.
As some one of the quantities Xj, X^, ..., X^, is not zero (for, thus
far, all these quantities are arbitrary), we shall take X,= l. In
order that this equation may hold, we assign definite values to
Xj, ..., A,; we write
B, + \B,+ ... + X,B, = 0,',
g, (^) + X.i'. (2) + . . . + X,5, (z) = G (^),
F. IV. 29
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450 DOUBLYPEllIODIO INTEGRALS [142.
and then, as Xg+i, ..., >.«, t^n remain arbitrary, wo have
forr = s + l, ...,K,. Moreover, on account of the composition of
Q (z), we have
and we had
Accordingly, the nmnber of integrals, which are doubly penodic
functions of the second kind and are associated with the multiple
root 0, of the fundamental equation ii{^) = 0, is
where % is the mimber vf elemental y divisors of ii(^) which are
powers of — 0,, "nd s n the number oj quantities
which do not vanish, so that < s < Ki
143. We now can indicate the total number of integrals,
which are doublyperiodic functions of the second kind.
Let 6, be a root of multiplicity X, of fi {6) = 0, and let it give
rise to n^ elementary divisors of fl (8) which are powers of ^ — ^j ;
and let s, be the number of quantities
in the preceding investigation which do not vanish, so that
<5i <«] €X,.
Let 0^, 03, ... be other multiple roots; and let X^, n^, s^; \^, jt,,
Sj ; . . . be the numbers for them, corresponding to X, , n, , s^ for 0^ ; so
that
X, + \ + \+... = m.
Then the number of integi'als, which are doublyperiodic functions
of the second kind, is
2 (1+M,,_s,.).
yGoosle
143.] OF THE SECOND KIND 451
In particular, if the roots of Cl(0) = O be all distinct from one
another, a fundamental system can be composed of m integrals,
each of which is a doublyperiodic function of the second kind;
the constant multipliers are the m roots of il (0) = 0, and the
corresponding quantities 0' derived from them, these quantities 0'
themselves satisfying the equation O.' (ff) = (I.
Moreover, the relation between the equations satisfied by
and Xi, ..., X™, and the equations satisfied by 0' and X,, ..., X™,
is reciprocal ; for each set can be constructed from the other as in
1 141. Hence, if either of the equations £1 (f) = and li' (0') =
has all its roots distinct from one another, there is no necessity to
take account of possible multiplicity of the roots of the other, so
far as the present purpose is concerned : the implication merely
is that one of the two multipliers has the same value for several
of the integrals.
Further, if 9 and 0' are two associated multipliers, each of
them arising as repeated roots of their respective equations, we
shall suppose, for the same reason as in the preceding case, that
the construction of the doublyperiodic functions of the second
kind is initially associated with that one of the two equations
which has the repeated root in the smaller multiplicity.
Multiple Roots of the Fundamental Equations and
Associated Interralh,
144. Wo have now to consider the form of the integrals
associated with a multiple root of II (8) = 0, the fundamental
equation for the period w ; and we assume that the correspond
ing root of li'(^') = is also multiple, to at least as great an
order of multiplicity. Denoting this root by 0, and the corre
sponding root of il' (6') = by 0', we know that there certainly
is one integral, which is doublyperiodic of the second kind and
has multipliers and 0': let it be denoted by .^, so that
<t>, {z + m) = 04,, (2), 0, {2 ^ m')  e'<i., {z).
Considering the integrals first in relation to the period m, we
know (§ 134) that the number of them associated with the
multiple root is equal to the order of multiplicity of 8: and
39—3
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452 INTEGBAL8 ASSOCIATED [144.
further, that this group of integrals is linearly equivalent to
subgroups of integrals of the form
Wj = 01 + 330, + 3£"0a + S^(j>„
the aggregate number of integrals in the various subgroups is
equal to the order of the multiplicity of 0, and each of the
functions is such that
^(^ + »)«+ (4
In these integrals, 0^ can have any added constant multiple of
01 ; also 05 can have any linear combination of constant multiples
of 0s and 01 ; and so on. All the functions 0, so changed, still
have the multiplier 8 for the period w.
Now u, has the multiplier ff for the period m'. The simplest
case arises when some other integral of the group, say %, also has
this multiplier $' for the period a>' : for then all the intervening
integrals have this multiplier for the period co'. What is neces
sary to secure this result is that, first,
,(., (» + »■) +(z + «,■) *, (, + «■) = »'!*, (^) + ^^, wi,
that is,
and therefore
■ ».(^ + »') ,., ».w
*,(»+»■) + " *«■
Secondly, we must have
.^, (2 + »') + 2 (« + «■) ^, (« + •>') + (^ + »•)• * (» + »')
9'l*.(.) + 2#,W + ^*(A,
which, in connection with the preceding equations, is satisfied if
05 (s + 0.') i 2«'0., (s + a.') H m'^0, (2 I «.') = 0<l>, (z),
that is, if
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144.]
WITH MULTIPLE
; ROC
Similaily, we
must have
4>,(z + (,
?!.
,*.(» + ■
'} + Sa"
,*.o
M^ + u
*.o
and so on.
•■) *W
Let ^(s) denote the usual "Weieratrassian ^function, with
periods oi and w'; and let 17, V denote the increments of ^(z) for
an increase of s hy the respective periods, so that we have
1701' — Jj'fii = + Stt*,
the sign being the same as that of the real part of a>' ? ita. Then,
if a function u{s) be defined by the equation
.(»+«
) = »W,
«(^ + «
■)«« + „'.
.(.+„■)*■«
^.(2)
"W.
that is, the function on the righthand side is periodic in m'.
Moreover, <J3^ and 0, have the same multiplier for q>, and u {z) is
periodic in w; hence the function on the righthand side is
periodic in w also. It thus is a doublyperiodic function of the
first kind ; denoting it by i/^.^, we have
so that ^)i^2 is a doublyperiodic function of the second kind,
with the same multipliers as 1^1, viz. 6 and &.
Similariy, we have
3 that the function on the righthand side manifestly is periodic
1 m' ; and it is periodic in co on account of the properties of m and
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454 INTEGRAI£ ASSOCIATED [144,
the functions i^. Denoting this doubly periodic function of the
first kind by i/fj, we have
And so on in succession. The group of integrals, in the case
id, can be represented in the form
where the functions ^i, ^^, ^s, Xi' ■■■ ^^^ doubly periodic functions
of the second kind with the multipliers 8 and 8', and
145. Returning now to the less simple case, when not more
than one of the integrals associated with the corresponding
multiple roots can be assumed to be doubly periodic of the second
kind, wc know that one integral certainly exists in the form of a
doubly periodic function of the seoond kind with the multipliers 6
and 8". Denoting it by </ii(s), we use it to replace some one of
the integrals, say fi(s:), in the fundamental system, which then
becomes
*(A /.(A .... /„('}■
We have
*(^ + »)=»*,(<:),
The fundamental equation for the period to ia
n(x)^\ ex, , , ..., =0;
and so f is a root of
of multiplicity less by one than its multiplicity for li {x) = 0.
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145.] WITH MULTIPLE ROOTS
Similarly, we have relations of the form
/, (3 +<c') = d^ 4,, (z) + d.^f.A^) + +d^ /™ (^X
the multiplier ff being a root of
D.{{x) = d.^x, d^, ..., d,„, 1 = 0,
of multiplicity less by one than its multiplicity for Xi' (x) = 0.
The coefficients cj and dij must satisfy conditions in order to
have
/.{. + «') + <.=/.l(^ + <.) + «'!.
for all values of k : these conditions are
dn(hi + 'ir3Cig + ... + drmCrm=^ Crldu + Cndis + ■■■ + Crmdms
= K.
say, with the limitations
Cn = 0, d„ = d'; c.s = 0, d,, = 0, it s>l.
Owing to the fact that ^ is a root of Oi(,k) = 0, quantities
*3, ..,, Km exist such that
^Ks^Cas + CssKal ... +C,„sK„,, (s = 8, ..., m),
^ = Cai + Cja/ira + ... +Cm2«m
Let
'^ = dss + dsiKi+ ... +d„,^K^,
iTfd^+dirK3+ ■■■ + datrK„. {r = 3, ..., m);
then
Sap = dir (ca + Csa K3 + ... + Cms «m)
+ '^3r (Csa +C33 Ka+. + Cm3 Km)
(^2 Car + "^ Csr + .  . + daiH Cmr
+ Kj (das Cffl. + rfa3 «»+■■• +'^sm Cmr)
+
+ «™ (<4.aCw + C^iaCar + ■ . . + (4™C™.)
; CjA + Car ffa + CirO4 + . . ■ + C™.(7™ ;
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MULTIPLE ROOTS AND [145.
holding for all values of r. Comparing with the oarlier equations
in c, we have
0,  ?»/<:,.,
for all values of r ; and thus the second set of equations is
a = rf^ + (;^Ks + ... + <^ffl3«m,
^/(:,.= (4. + 4,«:a+ ... +d,„K„, (r = 3, ...,m).
Eliminating the quantities rc, we have
so that 0' is the value of a.
Now consider the integral
X. W /. w + «./, (2) + . . . + «./.. (t).
We have
X. (» + »)/■ (2 +«) + «./,(» + ») + .•.+«,./,.<« + »)
= ".*« + "»(«),
say, and
» (' + »■) /■ {•:+•>') + «./. (« + »•)+... + «./„ (^ + »•)
= 6,*,W + «'X,«,
say. When Hj and 62 vanish, ^^ is doublyperiodic of the second
kind; but in the general case, a.^ and b^ are distinct from zero.
The property
x,l(^ + ») + .1»(^ + «') + »!
leads to no relation between 0,3 and b^.
If the multiplicity of (9 as a root of 12 (^) = and that of 6' as
a root of H' (^) = be greater than 2, so that and ^' are
multiple roots of ilj («) = 0, flj (x) = 0, we proceed as above. The
newly obtained integral j;, is used to modify the fundamental
system by replacing y^, say, so that the system consists of
A. X.. /. /»■
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145.] ASSOCIATED INTEGRALS 457
Then, in the same way as above, it is proved that an integral ;:^
exists such that
Since
»K^ + .) + »■) »!(« + »') + »),
we find, on aubstitation,
so that we may take
c^ — Xa^, di^Xb^,
where \ is any parameter. This parameter may clearly be
absorbed into j^, by taking Xa^V and also into a, and h^ by
division. Thus our integrals 0i, ')(_^, j^j are such that
Xa (s + » ) = 050, + a^x' + ^X3.
<^,(^ + a,') = '5'0„
And so on, until a number of integrals is obtained equal to the
leaser of the orders of multiplicity of 6 and 6'. Thus the next
integral is x* (say), where
X. (^ + « ) = a,0i + («j + ^o.) X= + a=Xs + ^X .
X, (s + <o') = b,<f>, + {h + \h) X, + b,x, + B'x. .
146. From these descriptive forms, we can proceed one stage
towards the construction of an analytical form of the integrals.
For this purpose, we introduce (as in § 144) the functions
where the doubtful sign is the same as that of the real part of
w' ^ i<it ; we have
Then the various integrals can be expressed as nonhomogeneous
polynomials in i( (s) and v (z), the coefficients of which are doubly
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458 ANALYTICAL FORM OF INTEGRALS [146.
periodic functions of the second kind, with and 0' for multipliers.
In particular, the integrals have the form
X, » = *,(.) + /.*■,(«).
x,W*'.W + r.*'.W+Wf.« + K.*',(4
and so on. The functions F are doubly periodic of the second kind
with factors 8 and &; /, is a polynomial of the first degree in u (s)
and v(z); /a and J^ are polynomials of the second degree in the
same quantities, having I^ as the aggregate of their terms of the
second degree ; /j is a polynomial of the third degree in the same
quantities, having I^ as the aggregate of its terms of the third
degree; and so on.
To prove this, we note in the first place that </>i {z} is a doubiy
periodic function of the second kind with the multipliers 6 and ff.
As for Xi (■2)1 ■'^e have
?6(g + <^) _ Xi(£) , ^
If therefore we take the function p,, (z), where
?.,(). J.
W + j1«W,
we have
*(» + .
iyM'*"
W
and therefore
^ »(» + «')
ft.(2>
P21O
is a doubly periodic function of the first kind. Let ^2(2) denote
the product of this function and 0i (z) ; then F^ (z) is doubly
periodic of the second kind with multipliers and 8', and we
have
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146.] ASSOCIATED WITH MUtTIPLE EOOT 459
Similarly, if
...«^{i(ty}.w.i()"^}«(4
we find
to be a doubly periodic function of the first kind. Let F^{z)
denote the product of this function and 0i(3); then Fi(z) is
doublyperiodic of the second kind with multipliers and &, and
wc have
X. (^)  ^. W + p. (') f. W + iiP=." (^)  r (')! * W
Similarly, after reduction,
X, (.) = F. W + p„ » ^, («) + (ip. W  ft, (2)} f. W
+ (iP.'Wp..W!*.W
where Fj (z) is a doublyperiodic function of the second kind with
multipliers 6 and $', pt, (s) is a polynomial in u and v of the first
degree, and p^ (2) is a polynomial (not homogeneous) in u and v
of the second degree.
And so on, in general : the theorem is thus established.
Construction of Integrals that are Uneeokm.
147. Further progress in the efifective determination of the
analytical forms of the integrals on the basis of the foregoing
properties is not possible in the general case. When particular
classes of limitations are imposed upon the coefficients in the
original differential equation, such progress might be possible :
but it frequently happens that some more special method loads
more directly to the solution.
The simplest case is that in which the equation possesses a
uniform integral, or in which the equation has several uniform
integrals: but, of course, the preceding investigations in §§ 141 —
146 apply to all equations of the type considered, whether they
have uniform integrals or not. When all the integrals are
uniform (and this can be determined independently by consider
ing their forms in the vicinity of the singularities), then the
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460 EQUATIONS HAVING [147.
doubly periodic functions of the second kind arising in the pre
ceding investigation are uniform functions of z; and a general
method of constructing such functions is known*. Instead, how
ever, of using the preceding results, it sometimes is more con
venient and more direct to infer the irreducible singularities of
the integrals from the differential equation itself. These are used
to construct an appropriate uniform doublyperiodic function of
the second kind; the remaining quantities needed for the precise
determination of the integral are then inferred by substituting
the expression in the differential equation.
Ea:. 1. Consider the equation t
with the usual notation for the Weierstrassian elliptic functions; q and ^ are
Constanta.
The only irreducible singularity that an integral can have is 2=0. The
iodicial equation for e=0 is
m(»I)(«2)6«=0,
the roots of which are 1, 0, 4; and the expansions that respectively corre
spond to the roots are easily proved to be
Wjll 1^(33^1^503!^ + ...,
Thus no It^arithniB are involved ; every integral is a uniform function of z,
being of the form. Aw^\Bw^\Ow^; and at least one integral of the equation
is thus a uniform doubly*periodic function of the second kind. We proceed
to its construction.
This doublyperiodio function of the second kind cannot be devoid of poles,
if it is to involve the first of the above integrals in its expreasion. (If it were
devoid of poles, it would alsoj be devoid of zeros in the finite part of the
plane : and then {I.e.') it could only be an exponential of the form e", which
is manifestly not a solution of oiu' equation.) It has one irreducible pole ; it
therefore has one irreducible zero in the finite part of the plane. Let the
latter be denoted by  a, which at present is unknown.
We now consider § the elementary function
" (■' + «) M
* T. J., g§ 137—139.
+ It ia a modified form of an eiiuation given by Picai'd, C'i'dle, t. kc, p. 290.
t T. F., i 139. g T. F., U.
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147.] UNIFOKM PERIODIC INTEGRALS 461
which has £=0 for an irreducible simple pole, and a for an iiTeducible aiinple
Kero; its oxpansion begins with s~i, and the function muat therefore agree
with the integral above obtained in the vicinity of 3 = 0. (The constants X
and a determine, or are determined by, the multipliers of the periodic function ;
but at present these are unknown, and so X and a must be detormined in
another maimer.) To expand the above function in powers of 3, we have
the Weieratrassian functions on the righthand aide being functions of a.
»(.)
J+(>.+0+i'{(»+0'B+i''»+0"3(i+f)»'rl
+£»+o*8(^+f)'f"ip+nf *■+•«)+•■■
This is to satisfy ttie dilferential equation, so tbat it must be of tlie form
Clearly J = 1, iJ = X + f, for this purpose: the value of P would be needed for
the complete expression, but we merely require X and « at present. Com
paring the coefficieutB, we thus have
^ = 1,
ax+C,
^— (x+O'f,
so that X and a are determined by the equations
(X+O'J'— 1
(x+o'3(x+f)t>p'.jfir
•V
x+f. _
where a is determined bj' the relation
Sjy + (3a>j,)J> + J3'a'J,.0.
The function on the lefthand side is a doublyperiodic function of tho first
kind: it has a single irreducible pole, which is at 3=0 and is of multiplicity
three. Hence it has three irreducible aeros, say a^, a^, a^; and their sum is
congruent to 0, bo that we may take
a^ + a^ + a^^O.
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462 EXAMPLES [147.
In general, u^, Oj, «j ure unequal, because a and /i are general constants; the
discussion of the critical conditions, that lead to equalities between a,, a^, a^t
and of the consequent modifications in the complete primitive, is left as an
K S W+ a„2g)(a,) ' ('1.2,3),
^^^)^
is an integral of the equation for each of the three values of r. Tho primitive
of the equation is
whore A^, A^, A^ are arbitrary constants.
Jix. 2. Obtain the relations which express the integrals w^, w^, v>^ of the
equation in the preceding example in terms of H";, W^, W.^; and determine
the multipliers of the integrals.
Ew. 3. Obtain the primitive of the equation
in the form
w=Ae<^'^ + Bf{z).
Ex. i. Verify that the primitive of the equation
da''^ dn.
y=^cos(nam^)+£sin{»iam:^).
Ex. 5. Prove that, if / be an odd function and J^ be an even function,
both doubly periodic in the same periods, the integrals of the equation
rfe= ds
oan bo expressed in tenns of JJ (z).
Hence (or otherwise) integrate the equation
cr ' r
J^x. S. Determine tbe relations among tlie constants (if any) in tlie
equation
»'"+(aSB»'+(y+(iJ)j)>.0,
in order that every integral of tlie equation aliould te uniform ; atid assuming
tile relations satisfied, shew that the equation has three integrals of the form
'(■+") ^
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147.] EXAMPLES 463
Ha. 7. Shew that the equation
has an integral of the form
^""^ ®(^) ^"^ "'
provided
3a + / = 3(l+i2).
and that it then has three integrals of that form. Obtain these integrals.
(Mitte^LefBer.)
Eu. 8. Obtain the integral of the equation
in the form
^ ■&(:.)
where the constants X and « ai'o given bj the equations
A(14(t=) + 3(A=i^8na«) = 0,
2X=6Xi"8n^B + 3X(HF)4i2sn«)cnQ.dncoAi=0.
Verify that, in general, three distinct integrals are thus obtained. (Picard.)
A'iB. 9. Prove that the equation
has an integral of the form
^ &(ic) '
provided
2a + S = 8(l+^)i
and that, if this relation he satisfied, it has fonr such int^rals. Obtain
them. (MittagLoffler.)
Em. 10. Verify that the equation
(m..,l'<.)g2.n>:«oxdn^J + 2jl2(I+i').n,. + 3f .■,•.).!
hiia an integral of the form
provided
an^'cisn^B ' afc2sn*K2(I+i=)sn«a + l '
and obtain the primitive.
Hence integrate the equation
where A is a constant
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464 lamp's [147.
Ew. 11. Discuss the equation
S + 2J,(.))*„J + ,, + ,p(.,,..„,
for those cases when every integral is a uniform function of z.
Ea:. 12. Shew that there are three sets of values of the constants o, and 6,
for which the equation
33(Ar"(»)+ir (»)+"?(")+»)?
admits aa an integral a uniform douhly periodic function ; and obtain the
integral (Math. Tripos, Part ii, 1997.)
Ex. 13. Prove that the equation
y'"2»(K + l)y'p(2)2w(ft + l)/^'(.)
where a ia an arbitrary constant and n is a positive integer, has a uniform
function of z for its complete primitive. (H^lphen.)
Ex. 14. Construct the equation which has
for its complete primitive and, for a properly determined value of /(a), is
devoid of the term in ^ . Likewise construct the equation which has
w={a,+agsn3 + 0!3cnj+aidns}/(j)
for its complete primitive, with the corresponding determination ai f{z) to
remove the term in ^ . In each case, the quantities a^, a^, %, a^ are to be
regarded as arbitrary constants. (Halphen.)
Ex. 15. Prove that the primitive of the equation
is a uniform function of z, when m is an integer multiple of 3 ; and discuss
the primitive, when the integer n is prime to 3. (Halphen.)
Lame's Equation.
148. One of the most important instances, in which a dif
ferential equation with uniform douhlyperiodic coefGcients has
a uniform doubly periodic function of the second kind for its
integral, ia Lamp's equation or, rather, the more general form of
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148.] EQUATION 465
Lamp's equation as discussed in the investigations of Hermite,
Halphen, and others. The form used by Hermite* is
where n is a positive integer and .S is a general constant; the
form used hy Halphenf is
with the same significance for n and B. We shall use the latter
form of equation ; it is selected for convenience and for its slightly
greater generality owing to the functional independence of the
The mode of discussion is the same for the two forms.
As we are concerned with the application of the general
theory!, rather than with the special properties of the functions
defined by Lamp's equation, only an outline of the solution of the
equation will be given here. The detailed developments, and
references to further memoirs, will be found in the authorities
just quoted.
It may bo not without interest to indicate how this form of
equation arises from the equation
da? dy' d^ '
characteristic of the potential in free space. When orthogonal
curvilinear coordinates a, /3, 7, as defined by three orthogonal
are used, then the equation becomes
do:[BC Sa>'^Sg\CAafil St\AB dy) •
* " Siir que <i e apjl at on deE foBctions elliptiques, " a separate repiitit
(1B85) Irom the C np He E s
+ Traite dts to ctio e Ipt q es t ch. xir.
J That is the theory of the un f rm doubly periodic functions of the second
kind which are ntegials of tlie differ nt al equation. It has been proved (g 54)
that, by an app 01 t anal in on the equation can be changed eo as to be of
Fuehsian type
I. IV. 30
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466 SOURCE OF [148.
where
«©■©■©■•
Choose, as the orthogonal surfaces, the three quadrics which are
coiifocal with a given ellipsoid ; and let \, fi, v be the roots of the
equation
ir} if ^° _ I
a cubic in 9. Then* we take
([2 + \=:^(a)  ^1, ¥ ■\\^f{oL)  e.,, c^ + \ =g>(a)  e^,
«' + ^ = ^(^)e„ 6^ + M = F(/3)~e.„ c^ + ^ = g)(^)e„
Now
jd\\^ /dXV /3XV
Kdcc) ^ \dyl ^ \dz}
 ^'^:
where
\ _ g? y s^
jiT" ^ (a' + Xf + (6" + X)" "'" (C + X)'
(Xy)( X »)
((•■ + X)(6" + X)(c" + X)
!>■■(«)
1 „, 1
Similarly
so that the equation for the potential becomes
* Greenhill, Pioc, ioiid. .Viif/f, Soc, t. xviii (18B7. p. 27S.
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148.] lamp's kquation 467
or, what ifi the same thing,
IP(«S>(7)lj„'^+l!'(7)?WlU+[S>Wp(ffll^ = 0.
For the purposes contemplated in the transformation, the quantity
V is the product of a function of a, a function of /3, and a function
of 7, or is an aggregate of such products ; and it is a uniform
function of its variables. Hence, writing
F=/(»)!,(/3);.(7),
where /, g, h denote uniform functions of their arguments, we
have
where w =f when z = a, w = g when z=^^, w = h when s — y, and
A, B are constants independent of a, /3, 7 : they must be such as
will, if possible, make v) a uniform function of its argument. The
only possible singularities of w are ^ = and points congruent
with z=0; hence, after the earlier investigations, we consider the
irreducible point 2=0. The form of the equation shews that it
will be an infinity of w; and thus it must be a pole, say of
order n, where n, is, & positive integer. Thus we have, in the
vicinity of the pole,
«; = J + ,^^,+ ... =^K(s),
where R (2) and iJi {z) are regular functions of z, such that
BA£)_
Hence, in the vicinity ni z — 0, we have
 j^ =— ^ — ^ — ' (1 + powers 01 z),
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468 INTEGRATION OF [14S.
and therefore
^.(n + l).
a limitation upon the form of the constant A. But there is no
limitation upon B, necessary for the existence of integrals of the
type indicated ; and therefore the differential equation may be
taken in the form as stated.
To obtain Hermite'a form, we write
, , _ «! — 63 _  ,^ J _e.^ — e^
f {s)  03  gnSy ' V ^(^1 ^3>. ■  g^ _ g^ •
as usual, and then take
y = cc + i'K' ;
the equation becomes
where B" is a constant.
14!). The method of solution of the equation is based upon
the knowledge that there is at least one integral in the form of a
doublyperiodic function of the second kind ; the limitations, that
have been imposed upon the equation, secure that this function is
uniform. Moreover, the integral has only one irreducible pole,
viz. at s = 0, and the pole is of order n.
There are two modes of using these results in order to
construct the integral.
By one of them, we use* the further property that a uniform
doubly periodic function of the second kind has as many irre
ducible zeros as it has irreducible poles, account being taken of
the orders of the points in each category. Accordingly, in the
present instance, the integral has n irreducible zeros : let them be
— rii, — «a, ., — <^ij Consider the uniform function
which is doubly periodic of the second kind; its (single) irre
ducible pole is of order n and is at s = 0; and it possesses the
* r. F., %% 139, 141.
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149.] lamp's equation 469
necessary « (unkiiown) irreducible zeros, so that it is of a suitable
form. We have
1 __
w dz
In order to simplify the righthand side, it is conveoient to take
and so
Hence
1 d?w 1 idwy , , o / ,
and therefore
1— _x+i I y'w^ p'w
li)(«,)yW(
.lii y'(n,)p'W p'(a, ) y'W
Tlie first term on the righthand side is equal to
To modify the second term, where the summation is for pairs of
unequal values of r and s, we have
li>'('V)y'W P' (»■)»'(»)
(>("»)pW ■ «>(«.)«'(»>
_ yW g.pWg.p'(»r)y'(''.)f'MI»'K) + t''(».)l
after easy reductions, where
i,.
?(«)!>(") PWPK))
. P' (»>) + P' (a. ) .
' P(»r)S)(».) '
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470 lamp's equation [149.
and thus the second turm in the expression for —^\j] becomes
»{,.i),,(.)+2(..i)_^ls.(«.,)+i_f'<>f^'l;i,,.
where the summation is for all values of 5 from 1 to w except only
s = r. Then
^  y(a..)yW {f,^ aJ,
mtWfM Ui )
Comparing this result with the differential equation under con
sideration, we naturally take
i' L,,  2\
for all values of r, that is,
»'M + f'('h) P'W + f'l". ) ^ _ 2^^
j>((i,)i)(ii,) J)(o,)>(o,)
f'W + y'W , y'(».) + t^(».) , _ 2x
fW>(«.) «>(".)«><».)
f'(°~) +»'(■■■) ^. g'W + y'W ^ _ 2^
«'(».)8'(»i) «'("•)(('(«!)
and (2"  1) ^ g* («.) + X^ = 5.
Adding the n former equations together, we have
= 2n\,
so that \ vanishes. Hence if the n quantities di, a^ a„ o.re
determined by the equations
y'W + y'W y'W + »''('■.) ^ _ „
(f'W + f'W , y'(».) + y'(a.) ^ ^ D
?(".)(»(»■) fWpW
{which are equivalent to only k — 1 independent equations, becaii^e
the sum of the n [efthand sides is zero) and by
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119.]
XNTEGKATED
thm
F{z).
a{z + a^)„{z + a.i...<,
»w
is an integral of Lamp's equation
i^ = «(« + l)^(2) + B.
w dz^ '
The equation remains unchanged when —2 is written for z;
hence F{—z) is also an integral. Save in the case when the
constants a are such that F{z') and F{z) are effectively the
same function, we have two independent integrals of the equa
tion, which therefore is completely solved.
150. Another method of arranging the necessary analysis is
as follows. Consider the equation
where F(z) is a doubly periodic function; by Picard's theorem
(§ 142), an integral (say Wi) is known to exist in the form of a
doublyperiodic function of the second kind. If then we write
_ 1 dwi
the quantity w is a doublyperiodic function of the iirst kind ; and
it satisfies the equation
The irreducible poles of w,, in their proper order, are known from
the singularities of the original equation ; let them be jt in
number, account being taken of multiplicity. Then each of them
is a pole of v, of the first order ; and the sum of their residues for
vis —n. The number* of irreducible zeros of w, is also n, account
being taken of multiplicity; each of them is a pole of v, of the
first order, and the sum of their resid\ies for v is +n.
We therefore construct a uniform doubly periodic function of
the first kind, having these poles, all simple, viz. the known poles
arising through the singularities of F, and the unknown poles
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472 EQUATIONS HAVING [150
arising through the zeros of Wi, taking care to have —n and +n
for the respective sums of the residues. The general expression
for such a function is known*: when substituted as a trial
function in the above equation, comparison of the results leads to
a determination of the constants.
As an illustration, consider the equation
1 iPw „ , , T.
The irreducible pole of g) (e), viz., 2 = 0, is the only irreducible
pole of Wi, and it is of the first degree. Accordingly, it is a
simple pole of v, with a residue —1. Further, there is (by the
preceding argument) only one other pole of ii : it is simple, and
has a residue + 1, As w is a doubly periodic function of the first
kind, an appropriate expression is
.i:(«c)fw + f(c) + i.,
say ; and b, c have to be determined by substituting in the
equation
* + .. = 2p(.) + a
Now
dv , , / i
and by the addition theorem for the ffunction, we have
'' + *j)Wj.(o)
Substituting, we have
if W + f(c) + b + b ^>+ ^'l) = 28, («) + B,
which must be satisfied identically. Accordingly,
6 = 0, ^(c) = B;
and thus, with a known value of c,
^ = j:(2c)r(5) + f(c),
* T. P., S 138.
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150.] DOUBLYPERIODIC COEFFICIENTS 473
SO that
There are two values o£ c, equal and opposite : the construction of
the primitive is immediate.
Ex. 1. Shew that two independent integrals of tiie equation
ill the case when 5 = ej, are given by
y(.),,(', (»>(.).,}»{(■(.+.)+.,.)!
and obtain the integrals in the cases, when B^e^, and B = e^, respectively.
K:'). 2. Obtain the primitive of the equation
(where q i.s constant), in the form
where
DiacuaM the solution in the three particular cases
<i = I+F, 1, lc\ (Ilermite,
Ex, 3. Shew that
satisfies the eqviation
if ^ ("i)' P ("2)1 ^ ("3) "^^ *hs roots of the oubic equation
and deduce the primitive. (Ilalphen.
Ex. 4. Shew that the primitive of the equation
can be expressed in finite form for appropriate values of the constant B it
the following cases : —
I. When *i is an even integer, =2»i, then either
where e^^, e are any two of the throe constants e^, ^j, e^ :
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474 EXAMPLES [1 50.
11. when )t is an odd integer, =2(1? — 1, then either
«'={PW«;.}*{c™_,iy"''(^)+...+C„},
where e^ is any one of the three constants e„ e^, e^.
Determine the number of sohitions of the specified kind in each of the
cases indicated. (Crawford.)
Ex. 5. Shew that an integral of the equation
where h is a coJiatant and m, n are integers, can he expressed in the form
_,»,(».,) a, (..,).. .8, fr....)
z zK)
in the usual notation of the thetaftinctions, ^,, z^, ..., ^mm hcing appropriate
constants.
Obtain the primitive. (M. Elliott.)
Bx. 6. Obtain the primitive of the equation
»4Ji%)<*«"<''''»*W«
Ea:, 7. Shew that there are two values of ij, for which the equation
i5i.+i,J>»2~r(') + t""S''».
where m is a constant, possesses an integral of the form
^ ,.t.;(..(«i').
,(■) ■
and, for each such value, obtain the primitive. (Bonoit.)
fie. 8. Shew that there are m+l values of k^,, for which the equation
where m is a constant and n a positive integer, possesses an integral of the
Prove abo that, if the righthand side of the differential equation be
increased by * (s), where * is a doublyperiodic function of the first kind
having all its poles simple, a corresponding theorem holds as regards the
integral, if it,, be properly determined. (Benoit.)
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150.] ALTERNATIVE PROCESS 475
Es. 9. Integrate the eqiiation
where n, n', n", n ate positive integers. (Darboui.)
151. The other mode of utilising the known properties of the
integral, when it is a uniform doublyperiodic function of the
second kind, is to obtain the actual expansion of the integral
in the vicinity of its irreducible pole and thence to construct its
functional expression in terms of the elementary function
»(,) ■
where a and \ are initially unknown constants. Some indication
of the process is given in Ex. 1, § 147; but a slightly different
form will be adopted for the present purpose. We take the
elementary function in the form
'{' +
where p and a are now to be regarded as the constants to be
determined. The expansion of this function in the vicinity of its
irreducible pole at s = is
+ sV {p'  6/>=F («)  ^99' («)  W («) + *?=} ^ + ■ ■ ■ ■
If, in the same vicinity, an integral of the diiferential equation
exists in the form
. . . + ' + (to + positive powers,
then we rnay take
where a comparison of expansions serves to determine the con
stants a and p. The integral thus is known.
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476 EXAMPLE [151,
An illustration will render the details clearer. In the case
when 91=2, the equation is
1 d'^J) „ , , n
Let
1 a,
w = ^ +  + (ti + ctjs + a,s^ + . ..
be substituted in the equation ; we find
a,= Q, ((,, = 0, a,0, ...
a, IB. a, = i,]Pi^g„...
so that
Manifestly, the form to take is
__dG_
dz '
and then eompaiing the two expansions, we have
ilp'p(«)liB.
/3(jy(»)p'(») = 0.
These equations give
BM27y.
3y'(a)
The former in general leads to two irreducible values of a ; the
latter uniquely determines p for each of these values of a.
Denoting the two values of a by a and — a, and writing
<r(«)<r(o)
the primitive of the differential equation is
... r<ie. . ..dc.
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151.] EXAMPLES 477
Bx, 1. Discuss tho integral of the equat.ioii
when a, as obtained in the preceding solution, has the values 0, <u, m', lu",
respectively.
Ex,. 2. Prove that an integral of the equation
in given b,
the constants p and a being given by the equations
Deduce the primitiv'
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CHAPTER X.
Equations having Algebraic Coefficients.
152. The differential equations, considered in the preceding
chapters, have had uniform functions of the independent variable
for their coefficients. We now proceed to consider (but only
briefly) some equations without this limitation : one of the most
important classes is constituted by those which have algebraic
functions of the variable as their coefficients. For this purpose,
let y denote an algebraic function of the independent variable
X, defined by the equation
where 1^ is a polynomial in x and y, and the equation is of genus
p. With this algebraic equation we associate the proper Eiemann
surfiwe of connectivity 2j) + 1.
We assume that the linear differential equation has uniform
functions of x and y for its coefficients, so that each of these is
a uniform function of position on the surface : and we write the
equation in the form
'""" + a, (a^, y) ;^ + a, (^, !/) ^:_~ + . . . + a,„ («^, ;/) ^^ = 0.
rf^™^"=^^^^rf^'
„=Jf
:, y) dx
the exponential in the factor of u on the righthand side being an
Abelian integral ; then the equation for w is
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152.] ALGEBRAIC COEFFICIENTS 479
devoid of the derivative of order m — l; all the coefficienta
Pa, ■. Pm ^>^^ algebraic functions of x, and are uniform functions
of X and y. This is the form of equation which will be discussed.
Let (iCj, jio) denote any position on the surface, which is not a
singular point on the surface and in the vicinity of which each of
the coefficients P is regular. Then an integral exists, which is
regular everywhere over a domain in the surface, and is uniquely
determined by the assignment of arbitrary values to iv and to its
first m — 1 derivatives at (ic,,. po) I" f*ct, all the results relating
to the synectic integrals of an equation with uniform coefficients
hold for the present equation in the domain of (x^, y^.
Next, let account be taken of the singularities of the equation
■>r = and of the associated surface. As these affect all the
coefficients of all differential equations of the class considered,
and thus afford no relative discrimination among the functions
defined by those equations, we shall assume them simplified as
much as possible before proceeding to consider the properties of
the functions. Accordingly, we shall suppose that, if the equa
tion "It = {or the Riemann surface associated with it) possesses
a complicated singularity, it is resolved* into its simplest form
by means of birational transformations, so that we may write
where ff and h are uniform functions which, in connection with
^ = 0, admit of uniform expressions for f and 17 in terms of
«— e and y—f, and are such that ^=0, »j = is an ordinary
position on the transformed Riemann surface. The positions on
the surface, that have to be considered in connection with the
differential equation, are now ordinary positions : and therefore, in
dealing with the theory of the equation, no generality is lost if we
assume that the singularities of the equation 1^=0 and of the
Riemann surface are ordinary positions for the integrals. (Of course,
in any particular example, it may happen that a multiple point on
the curve ^ = 0, or a branchpoint of the associated surface, is
definitely a singularity of the equation. In order to discuss the
nature of the integrals in the vicinity of such a point, we takef
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480 FUKDAMENTAI, SYSTEM [1,52.
where ]) and q are integers, and S is a holomorphic function of
its argument that does not vanish when f=0; and then we
investigate the character of the integrals in the vicinity of
fO.)
Lastly, let (a, b) denote a position on the Riemann surface
(being a pair of values given by the differential equation) such
that the coefficients of the equation are not regular in the
immediate vicinity of (a, b) ; after the preceding explanations, we
may assume that ?/ — 6 is a holomorphic function of x — a in the
immediate vicinity of the position. The character of the integrals
in that region is determined, after substitution oi y — h in terms
of x — a, in association with an indicial equation ; and the general
processes of the theory, in the case of differential equations with
uniform coefficients, are applicable to the integrals in the vicinity
of {a, b). As in that earlier theory, we have a fundamental system
of integrals existing at any ordinary position on the surface, the
system being composed of m linearly independent members.
Continuation of these integrals is possible : and by taking all
admissible paths iirom one ordinary position to any other ordinary
position (care being taken to avoid the actual singularities), and
assuming an arbitrary set of initial values at the first point, we
shall obtain all possible integrals at the second point. Similarly,
by taking all possible closed paths on the Eiemann surface,
which begin at an ordinary point (.x^, ya) and return to it, we
obtain new integrals at the end of the path ; and each of these
integrals is linearly expressible in terms of the members of the
initial fundamental system.
A Fundamental System op Integrals, and the
Fundamental Equation.
153. Let Wi, Wi, ,.., Wm denote a fundamental system at
an ordinary position («o, y„); and let the variable of position
describe a closed path on the surface returning to (a:„, 7/0), this
closed path being chosen so as to include the singularity (a, h)
but no other singularity of the differential equation. Suppose
that the effect upon the fundamental system, caused by this
variation of the variable of position, is to change it into
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153.]
OF ISTEGRALS
481
w^, w^, ..., Wm : then, as in the case of uniform coefficients, the
latter set also constitute a fundamental system, and the two
systems are related by the equations
Mj/= 2 a^^w^,
(m = i,
ni\
where the determinant of the coefficients a is different from zero.
This determinant is (as in § 14) equal to unity. For let A denote
the determinant of the fundamental system
and let A' denote the same determinant in relation to the
fundamental system w' ; then, if A denote the determinant of
the coefficients a^^, we have
A' = ^A.
Now, because the term involving the {m
is absent from the differential equation, wi
— l)th derivative of w
have, as in § 14,
where C is a constant. Let the function A, which is equal to C
in the vicinity of (a:^, y^, be traced along the closed path which
the variable of position describes on its return to {xa, y^\ it is
steadily constant, and its final value is A', so that
^ = 1.
Further, as in the cases when the coefficients are uniform
functions of the independent variable, it is possible to choose a
linear combination v of the members of the fundamental system
such that, if if denote the value of v obtained by making the
variable of position describe the afoi'esaid closed path, we have
31
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482 FUKDAMENTAI. KQUATION [153.
The multiplier ^ is a root of the equation
«2i , Hsa — ^ flam
= 1 + I,B + 1,0' + ... + i^,^' + ( l)"^"" = 0.
This equation is independent of the choice of the fundamental
system, so that its coefficients may be regarded as invariants of
the linear substitution, which the fundamental system undergoes
in the description of the closed path round («, b).
154. If some, or if all, of the integrals in the vicinity of (a, b)
are regular in the sense of § 29, then an indicial equation for the
singularity exists; and if p be a root of this equation for an
integral with a multiplier 8, then
If no one of the integrals is regular, there is no valid indicial
equation. In the first case, the general character of an integral is
determined by the value of p : and the explicit form is obtained
by substituting an expression of the appropriate chai'acter so as to
determine the coefficients. In the second case, various methods*
for obtaining the value of S have been suggested, by Fuchsf,
Hamburgerj, and Poincare§; the most general is the method of
infinite determinants, due to Hill and von Koch, and expounded
in Chapter Vili.
Without entering upon details, it may be said briefly that
many of the properties of linear differential equations having
algebraic coefficients can be treated by processes that, except as
to greater complexity in the mere analysis, are the same as for
equations with uniform coefficients. It therefore seems un
necessary to discuss them at any length, as they would lead to
what is substantially a repetition of a discussion already effected
for less complicated equations.
■ Seo g 127.
t Creiie, t. lxsy (1873), pp. 177—223.
J CrelU, t. LKXKiii (1877), pp. 185310.
§ Acta Math., t. iv (1881), pp. 208 et aeq.
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154.] EXAMPLE 483
A systematic discussinn of equations having algebraic coeffi
cients and development of many of their chaiacteristie properties
will be found in a series of memoirs by Thom6*.
Ex. 1. Consider the equation
where the variable y is dciined by the relation
' +y=i
and a, a, b, c are Luiistant^
The position, at infiiiitj i& a singuliiitj of the difleiential equation in
©aoh of the two sheets ot tlie Eiemann suifice The integrals aie regular
in that vicinity in one oheet ^nd the exponents to which thej belong ire the
roots of
provided o + 6i is not aero ; but, if a + bi=0, the integrals are irregular at
infinity in that sheet. Similarly, they are r^ular in the vioinity of infinity
in the other sheet, and the exponents to which they belong are the roots of
'<'+'>+5^K).'''
provided a — bi is not zero; but, if a — iii = 0, the integrals are irregular at
infinity in that sheet.
The other singularities of the equation are given by
ax+byVC=0\
When these are distinct hma one another, let them be denoted by ; =cos fl,
y = sm6; x=aoa(j>, y=3mi^ The integrals are regular in the viuimty of
each position; and the leajectue mdiciil equations are
''"■^" + („ S°c.t«/ °
'''"■^ (»i°c.« <°'
When the two singularities coincide, let the common position be denoted by
a:=c<m\lf, y — sirt'jr ; and then
In the vicinity, we have
ar=cosJ,+a
, = .intcot^i_^!^ + jC^...,
* Crelle, I. cxv (1895). pp, 33—33, 119—149 ; ib., t. csix (1898), pp. 131—147;
ib., t. oixr (1900), pp. 1—39; ib., t. cslsii (1900), pp. 1—39.
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484 EXAMPLES OF EQUATIONS HAVING [154.
so that the equation is
The integrals are not r^ular ; but the equation may have one normal
integral, and can even have two normal integrals, of the type
aUin V !
■where / is ft polynomial in ^. The forms, and the conditions necessary to
significance, can be obtained as in §§ 86 — 87.
Ex. 3. Discuss in the same way the singularities of the same differential
equation, when the irrational quantity y is given by the respective relations
(i) ^+f = \,
(ii) /4i^yja^^3.
Ex. 3. Let w, and a, denote a fundamental system, of the equation in
Es. 1, for 3' = (1— a^)^ ; and let «, and r^ denote a fundamental system of the
same equation for y—(la^)*. Shew tliat the linear equation of the fourth
order, which has Mj, it^, Vj, v^ as its integrals, has rational functiona of x for
its coeffioieiits ; and obtain them.
Ex. 4. The equation
has its primitive in tlie form
It is natural to inquire whether an equation
£^^,.
can have an integral of the type
where nr (a:, y) is a rational function of a: and y. A general method for such
aa inquiry has been given by Appell*, thoi^h it is not can'ied to a complete
issue as regards detail ; it will be sufficiently illustrated by means of the
equation
iP"W_ a x^^y _
where a?^y^ = \, it being required to find under what conditions, if any, the
equation can have an integral of the form
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154.] ALGEBRAIC COEFFICIENTS
where nr (,«, y) m s, rational function of x and y. Since
We assume that each of the quantities a+flt, a+hi, is different from nero.
By adopting the method in the precediug Ex. 1, the integrals of the equation
in w are easily aeen to b« regular in the vioinity of a; = co , so that they have
the form
where ni] is a holomorphic function for large values of x, not vanishing
when :e=co ; and thus
Substituting in the equation for or, we have
Now the infinities of w are included among the points
(i) :e= tc , which has juafc been considered ; there are two jwsaible
values of X in each sheet :
(ii) y = 0, with .r = l, ^=1, which are the branchpoints of the
surface :
(iii) ax\hy = <i, in each sheet.
Moreover, the zeros of w are uuknowti from the differential equation : but
they must bo considered, becaiise each of thetu gives a pole of la. Let such
the number of suoh points being unknowiL All these points, whether
infinities of w or zeros of w, can be singularities of m.
As regards the branchpoints (ii), we may take
y = 'i, ^=IiiH...,
in the vicinity of 1, 0, where jj is small ; and then
so far as the governing term in ct is concerned. If tiiis be
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486 EQUATION OF SECOND ORDEIl [154.
where ft>0, then
n+2=2n, A'^+nA=0.
Thus n=2; and we can have ^= — 2, or ^ = 0, as possible values,
Similarij for the vicinity of — 1, 0.
Next, at the two points (iii), where as+iri/=0, we have
^^^sinvf', y = coii\jr, a tan 'jr =  b.
Then, in the vicinity, wo take
,K=^ain.;^ + , y = oosi.^Uni^+.,.,
Thus the equation is
d^ (.1.8..) »1
de*' («■+»•)' e'
su far as the gaverning term in cf is concerned. If this governing term be
T .(»■+» ) 1
' ' (»■+!.•)■ ■
Thus there are two possible values of rr at each of the two points.
Lastly, as regards a point such as ,i;=/in the set (iv), it is easy to see that,
if the governing term in ot be
B
(^ /')•"
2n = n + ^, B'i = nB ;
that 18, 11=1, and either B = \, 5=0, are possible values. This holds for
every such point a:=f and in each sheet.
Our required function luix, y), if it exists, is to be a rational function of
X and j/t and we have obtained all the singularities that, in any circumstances,
it might possess. We accordingly must take some combination of the possible
iiifinitiea, which are
j; = oo , with any of the values of X,
^±l,y=0, with either A = 2,oi A=0,
a3: + by = Qi, with any of the values of a,
^=/, with 5=1, or 50.
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154.] HAVING ALGEBRAIC COEFFICIENTS 487
A possible form is clearly
a
where (7 is a constant. We have (if this be admissible)
a + bi '
from the first of the possible infinities : we take A = from the second ; then
<'='°"?'
from the third ; and we take 3=0 from t!ie fourth. Hence we muat have
abi
for some possible values of X and of a : that is,
{gyp)'=^T{'<.>s>?'«}'}
the signs being at oiir disposal. Thi.s leads to a single value of ft via.
and the condition ia satisfied by taking the negative sign on both siiies.
We then have
so that, with the above value of 3, an integral of the equation
d^^ 2/iaxibyy
is given by
Actual evaluation of the integral in the exponential can easily be effected.
se, it would have been possible to discviss the particular equation
by taking
=l+(i' ^~i+(a'
with ( as the new independent variable ; for the algebraic relation is of genua
zero, and therefore* the variables can be expressed as rational functions of a
new parameter. The new form of equation would then have uniform coeffi
cients. But the foregoing method, that has been adopted, ia possible for an
equation ijr {x, y) = of any genua.
* T. F., g 247.
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INTRODUCTION OF [155,
Association with Auxomokphic Functions.
165. It is manifest that some of the complexity in the
analysis associated with the construction of integrals, either in
general or in the vicinity of particular points, would be removed,
if the equation could be changed so that, in its new form, its
coefficients are uniform functions of the independent variable.
This change would be secured, if both the variables x and y in
the relation
were expressed as uniform functions of a new variable z.
Now it is known* that, when the genus of this relation is zero,
both X and y can be expressed aa rational functions of a new
variable z, which itself is a rational function of x and y. moreover,
the expressions contain (explicitly or inaplicitly) three arbitrary
parameters, which may be used to simplify the form of the
resulting equation. Againf, when the genus of the relation is
unity, both a: and y can be expressed as uniform doubly periodic
functions of a new variable z, while tg(z) and ^ {z) are rational
functions of x and y ; moreover, the expressions contain (explicitly
or implicitly) one arbitrary parameter, which again may be used
to simplify the form of the resulting equation. And, in each case,
definite processes are known by which the formal expressions of x
and y, in terms of the new variable, can actually be obtained.
When the genus of the algebraical relation
^{!e,y) = ^
is greater than unity, a corresponding transformation is possible
by means of automorphic functions : not merely so, but such a
transformation can be efl'ected in an unlimited number of ways.
Further, it is possible to choose transformations that simplify the
properties of the integrals of the diflerential equations to which
they are applied. But, down to the present time, the instances
in which the complete formal expressions of x and y have been
obtained, and the application to the differential equations has
been made, are comparatively rare. The results that have been
* T. F., % 247.
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155.] AUTOMORPHIC FUNCTIONS 489
established are of the nature of existence theorems. It is true
that indications for the construction of formal expressions are
given ; but the detailed analysis required to carry out the indica
tions is of so elaborate a charactet that it may fairly be said to be
incomplete. The subject presents great, if difficult, opportunities
for research in its present stage.
A brief account, based mainly on the work* of Poincar^, is ail
that will be given here. References to the investigations of Klein
and others in the region of automorphic functions will be found
elsewheref.
The main properties of infinite discontinuous groups and of
functions, which are automorphic for the substitutions of the
groups, will be regarded as known. It is convenient to associate
with any group a region of variation of the variable which is a
fundamental region ; and for the sake of simplicity in the following
explanations, it will be assumed that this region is such that,
when the substitutions are applied to it in turn, the whole plane
is covered once, and once only. Further, also for the sake of
simplicity, it will be assumed that the axis of real quantities in
the plane is conserved by the sufetitutions of the group. There
are corresponding investigations, which establish the results when
these assumptions are not made; but, as already indicated, the
results are mainly of the nature of existencetheorems and cannot
be regarded as possessing any final form, so that the kind of con
sideration adduced will be sufficiently illustrated by dealing with
the simplest cases. In order to deal with the most general cases,
it is nece^ary to utilise the theory of automorphic functions in all
its generality ; yet the subject still is merely in a stage of growth,
being far from its complete development^,
156. It is known § that, if x and y be two uniform
functions of a variable z, which are automorphic for an infinite
* This work ia beet esponnded in his five valuable memoirs in Acta Matheviatica,
t. 1 (1882), pp. 1—63, 193—294. ib., t. in (1883), pp. 49—99, ib.. t. iv (1884),
pp. 201—312, ib., t. V (1884), pp. 209—278.
+ T. F., chapters isi, ixii.
t The most oonseeotive account of the subject is to be found in Frieke und
Klein's VorUmngen ii. d. TkeorU d, mttmiiorphen Functionen (Leipeig, Teahner;
vol, I, 1897; vol. 11, part i, 1901).
§ T. F., § S09.
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'490 AUTOMORPHIC [156.
discontinuous group of substitutions effected on z, then some
algebraic relation
+(«,j)o
subsists between them. Conversely, if this algebraic equation be
given, it is desirable to express the variables x and y as uniform
automorphic functions of a new variable z. For this purpose, we
note that for general values of x, the variable i/ is a uniform
analytic function* of m ; but there are special values of x, being
the branch point 8, at and near which y ceases to be uniform.
Now suppose that x can be expressed as a uniform automorphic
function of s, say
the fundamental polygon being such that the branchpoint values
of X cori'espond to its comers (or to some of them), which include
all the essential singularities of the uniform function /(s). Then,
when substitution is made in the above relation, it becomes an
equation defining ?/ as a function of ^ ; so long as z varies within
the poiygona! region, y does not approach those values where it
ceases to be uniform, for they are given only by the corners of the
polygon. Hence y becomes a uniform functiont* of s ; and as a; is
automorphic for the group of the polygon, it is at once seen that
y also is automorphic for that group.
Further, suppose that at the same time there is given a linear
differential equation of any order, in which the coefficients are
rational functions of sc and y. In addition to the branchpoints
which may be singularities of the equation, it naay have a limited
number of other singularities. Let such a singularity be x = a,
y — h, where of course 'if {a, b) = 0: for the moment, the question
of the regularity of the integrals in the vicinity is not raised. If
the polygon is constructed, so that x = a corresponds to one of its
corners which is an essential singularity of the group, then that
corner is an essential singularity of /(a). Hence, when the
differential equation is transformed so that z becomes the in
dependent variable, the original singularities no longer occur so
long as 3 is restricted to variation within the fundamental polygon :
they can occur only for the special values at the corresponding
If, further, the function f(s) is such that no special
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156.] FUNCTIONS 491
singularities for values of s arc introduced for values of w that are
ordinary points of the equation, which will be the case ii f'{z)
does not vanish within the polygon, then all the values of z within
the polygon are ordinary poiuts of the equation, and all the
integrals are synectic everywhere within the polygon. The
singularities have been transferred to the boundary of the sregion;
and thus the variables x and y, as well as all the integrals of the
given linear differential equation which has rational functions of x
and y for its coefficients, can be expressed as uniform functions of
z within the region of its variation.
AuTOMouPHic Functions and Conformal Bepbesenta.tion.
157. The relation between the variable s and the function
x=f{s) can be considered in two different ways, the analytical
expression of the significance being the same for the two ways.
In the first place, the relation can be regarded as one of
conformal representation. Assuming for the sake of simplicity
that all the singular values of x are real, consider the problem*
of representing the upper half of the 3;plane bounded by the axis
of real quantities conformally upon a polygon in the splane,
bounded by circular arcs and having m sides : this conformal
representation is known to he possible. If its expression be
then f'{z) must not become zero or infinite anywhere within the
polygon, that is, for any finite values of x; for otherwise, the
magnification would be zero or infinite there, a result that is
excluded save at possible singularities on the boundary.
It is manifest that the representation remains substantially
the same, if the splane be subjected to any homograpiiic trans
formation
where a'd' — h'd — \\ for this will merely change the polygon
bounded by circular arcs into another polygon similarly bounded.
* T. F., % 271.
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492 CONFORMAL REPRESENTATION AND [157.
Hence, in constructing the function for the conformal representa
tion, account nmst be taken of this possibility; and therefore, as
{'.
4 = B,«'l,
where
{'.']
is the Schwar
ziai
1 derivative,
we
construct thi
tion {z,
4
We hare'
k«i=i5;
1
^ + s^
o°
. il(x),
(S
0) + *:.
say, where the summation on the righthand side extends over all
the singular values a of a;; the interna! angle of the spolygon at
the corner homologous with a is arr, and the coefficients A^ are
real quantities. If oo is an ordinary value of x, so that no angular
point of the polygon is its homologuo, then
= tA„
= XaA + lS(laa
= 2aU»+Sa(l~a^).
li' oD is a singular value of x, whicii has an angular point of
the polygon as its homologue, with the internal angle equal to ictt,
then
the summations being over all the finite singular values of x.
The number of constants is sufficient for the representation.
In the case when oo is not the homologue of an angular point of
the polygon, we have m constants a,, m constants a, and m con
stants A„, subjected to three relations as above; as all these
constants are real, there are 3m — 3 independent constants.
But, if
_ a'X + b"
^ c"'X + d" •
where a"d" ~ ^"c" = 1 and the constants a", b", c", d" are real,
then the upper half of the icplane is transformed into itself;
hence the m constants a are effectively only m — 3 in number,
and thus the constants in / («) are equivalent to 3m — 6 inde
pendent constants, which can be used to make a solution determ
■ The whole iovestigation is due to Schwarn; see T. F., g 371.
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157.] AUTOMORPHIC FUNCTIONS 493
inate. On the other hand, to determine the polygon, 3m constants
are needed, viz. two coordinates for each of the m corners and a
radius for each arc : but these are subject to a reduction by 6, for
the representation is determinate subject to a transformation
c'^+d"
where a'd' — b'c' = 1, and the constants a', b', c', d' are complex, so
that there are six real parameters undetermined. The number of
available constants is therefore sufficient for the number of condi
tions that must be s
In the case when =0 is the homologue of an angular point, we
have ni — 1 constants a, m constants a, and m constants A„, sub
jected to two relations as above; as ail these constants are real,
they are equivalent to 3m — 3 independent constants. The re
mainder of the argument is the same as before ; and we infer that
the number of constants is sufficient to satisfy the number of
conditions for the conform al representation.
It need hardly be pointed out that, thus far, the polygon
bounded by circular arcs is any polygon whatever; it has been
taken arbitrarily, and it does not necessarily satisfy the conditions
of being a fundamental region suited for the construction of auto
morphic functions.
158. That polygons can be drawn in the aplane, suited to
the construction of autoraorphic functions in connection with a
given algebraic relation i/r (x, y) = 0, may be seen as follows. For
simplicity, let the polygon be of the first family*, and let it
have 2n. edges arranged in n conjugate pairs ; and suppose that q
is the number of cycles of its corners, each cycle being closed.
The genus p of the group is given by
2p = m i 1  g.
When the surface included by the polygon is deformed and
stretchetl in such a manner that conjugate edges are made to
coincide by the coincidence of homologous points, then for each
cycle in the polygon there is a single position on the closed
" T. F., %% 203, 293.
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494 FUNDAMENTAL REGION [158.
surface obtained by the deformation. This closed surface corre
sponds* to the Riemann surface for the equation
^}r {x, y) = 0,
which also is of genus p ; and thus there are q positions on the
surface. ea<;h associated with one of the q cycles. Each such
position requires a couple of real parameters to define it; and
thus we have 2§ real parameters. Equations, which are biration
ally transformable into one another, are not regarded as inde
pendent r and therefore the effective number of constants in
^ {^' y) = to be taken into account is 'ip — 3, being the number"!"
of classmoduli which are invariantive under birational transform
ation. Each of these is complex, so that the number of real
parameters thus arising is 6p — f). We therefore have to provide
for 6p — 6 + 25 real parameters, by means of the polygon.
In order that the polygon may be properly associated with a
Fuchaian group, it must satisfy certain conditions. Its sides must
be arcs of circles, the centres of which lie in the axis of real
quantities. As it has 2n sides, we therefore require 2ji centres on
that axis and tn radii, making 4ft real quantities in all ; but three
of the centres may be taken arbitrarily, for the polygon now
under consideration is substantially unaffected by a transforma
tion
V ' cz + dl '
where a, b, c, d are real ; so that the total number of real quanti
ties necessary is effectively 4« — 3. They are, however, not suffi
cient of themselves to specify an appropriate polygon 1 for
conjugate sides must be congruent, a property that imposes one
condition for each pair of edges, and therefore n conditions in all :
and the sum of the angles in a cycle must be a submultiple of 2n;
so that q conditions in all are thus imposed. Hence the total
number of real quantities necessary is
= 4m — S — n — q
= Sn. ^~q
= 6p~6 + 2q,
in effect, the same as the number of real parameters given.
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159.] FUCHSIAN EQUATIONS 495
AUTOMORPHic Functions and Linear Equations oB' the
Second Order : Fuchsian Equations.
159. In the second place, the variable s, and the automorphic
functions x and y, can be associated with a Unear differential
equation of the second order. Let
then it is easy to verify that
v^ da? «s &3? IV ^ '
where (*', z\ is the Schwarzian derivative of as with regard to z,
and 1^ = dxjd,z. It is a known property* that, if *■ is an auto
morphic function of z, tben the function
is automorphic for the same group; hence it can ]
rationally in terms of a; and y, where
Denotiog its value by — 21, where / is a rational function of
X and y, which may be a rational function of x alone, we have
V] and Vj as linearly independent integrals of the equation
,„ + /jf = ;
the quantity z is the quotient of the two integrals.
The analytical relation is effectively the same as before ;
for if
{z,x] = 2I,
we knowf that z is the quotient of two integrals of
n Difereiitial Kquatioi
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496 AUTOMOKPHIC FUNCTIONS AND [159.
Moreover,
90 that the results agree in form. The difference is that, regard
ing the relation as a problem of conformal representation, we
have been able to calculate the value of / in gi'eater detail than
in the alternative mode of regarding the relation: but the con
siderations adduced in connection with the differential equation
have been of only the most general character, and have not
permitted any discussion of the form of /.
When an equation of the form
da?
is given, where / is a rational function of x, or a rational function
of two variables x and y, connected by an algebraic equation
t(!l!.j)0,
it may happen that x and y are uniform fuQctions of z, the
quotient of two integrals of the differentia! equation. But these
uniform functions are not necessaiily, nor even generally, auto
morphic for a group of substitutions of s. Judging from the result
of the consideration of the question as a problem of conformal
representation, we should be led to expect that the constants,
which survive in / after the conditions for uniformity are satisfied,
might be determinable so that the uniform functions of z are
automorphic. When this determination is effected, the equation
is called* Fuchsian by Poincar^j if the group be Fuchsian.
160. We proceed to consider more particularly the properties
of the equation
,„/)i = fl,
in relation to the qiiotient of its integrals. Let jc = a, y = h be a
singularity of the equation, where ■^{a,b) = 0; and let
Limit [{x — ayi]it=a = p,
so that the indicial equation for a is
n(nl) + p = 0.
' Acta Math., t. IV, p. 323.
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160.] FUCHSIAN EQUATIONS 497
Let n, and n^ be its roots, when they are unequal ; then two
integrals of the equation are of the form
and BO
If atr be the internal angle of the jrpolygon at the angular point
homologous with a, we must have
and therefore
that is,
so that
— 4p =
1a
the remaining terms being of index higher than — 2.
This is valid, if a is not zero. When a is zero, ao that «i = %
and therefore p — i, the integrals of the equation are
jjj = (3;([)«i[[l + ...} log (fl! «) + powers oix — a],
and so, in the immediate vicinity of a, we have
2 = — = log («! — »)+ powers ;
and then
the remaining terms again being of index higher than — 2.
The quantity a, in terms of which the leading fraction in Z is
expressed, depends upon the character of the singularity at (a, b).
If the latter denote a singular combination of values for the
equation
then it is known* that the variables x and y can be expressed in
the form
' T. F., § 97.
T. iv. 32
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498 FUCHSIAN [160.
where S(^) is a regular function of f. which does not vanish when
f = 0, and the expressions are valid in the immediate vicinity of
the position. Let r be the least common multiple of p and q, and
write
then in that vicinity, we have
{xar = z+...,
SO that both x and y are uniform functions of s in the vicinity.
The commonest instance occurs, when {a, b) is a simple branch
point ; we then have
so that a = 4,
If (a, b) he a singularity of some given differential equation of
any order, say
where i/r (x, y) = 0, three cases arise.
Firstly, let all the integrals be free from logarithms, and let all
the exponents to which the members of a fundamental system of
integi'als (supposed regular) belong be commensurable ; then they
are integer multiples of a quantity k~^, and we take
..l. (.„, = ......
In that case, any integral is of the form
= (a; — o)* R(a; — a)
''RC),
SO that the integrals of the equation, as wel! as the variables
w and y, become uniform functions of z in the vicinity of 2 = 0.
Secondly, let the integrals (still supposed regular) of the
fundamental system belong to exponents some of which at least
are not commensurable quantities. We take
ir = log(a!a)powers;
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becomes
integral of the form
ix~arR(a:a),
i.e., a uniform function of s, valid for large values of \z\ : and this
uniformity is maintained whether /j, is commensurable or not.
Thirdly, let x = a be an essential singularity of one or more of
the integrals, supposed irregular there. As in the laat case, we
take
2 = log (ai ~a) + powers ;
the integral may or may not become uniform for large values
of U.
In the last two cases, if the expression for a: in terms of s, say
be automorphic, then 2 = os is an essential singularity of the
function f{s) ; and then, when z varies within the polygonal
region, w does not approach the value a for which the integrals of
the equation cease to be regular. Within the region, the integrals
are unifoim. It is to be noted that the relation, adopted in the
second case and the third case, woufd be effective in the first case
also, so far as securing uniformity ; but the converse does not
hold. The relation which, as seen above, corresponds to the
vicinity of an angular point of the polygon where the sides touch,
is the most generally applicable of all : the form of relation, corre
sponding to the first case, is applicable only under the somewhat
restricted conditions of that case.
161. These conditions and limitations affect the quantity /
in the equation
for they determine the leading coefficient in its expansion near
any of its poles ; but, in general, they do not determine / com
pletely. On the other hand, we so far have only secured the
uniformity in character of the functional expression of x in terms
of z: the automorphic property of the functional expression has
not been secured. The latter is effected by the proper assign
ment of the remaining parameters in /.
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500 CONSTRUCTION OF [161,
Aa a special instance, take the case in which the genus of the
group and of the permanent equatii>n is zero ; so that, if the
polygon has 2m edges, the number of cycles q is given by
, = » + !.
Taking the angulai' points in order as A„ A^, ...,A^a, and making
the sides
A,A, I A,A, ] ] A,^,A^ ] An ^„+,)
A,aJ' A,„A^Jy A„+,A„J' A„+,A^J
to be conjugate pairs, the necessary n+1 cycloa are
A,] A„A,^; A,4™i; ■; A,„A„+,; A„+„
To define the polygon of 2n circular arcs, which have their
centres on the axis of real quantities, we require the 4in coordi
nates of the angular points ; but these effectively are only 4k — 3
quantities, because the ^plane is determinate, subject only to a
transformation
/ as + b\
V ' cz + d)'
where a, b, c, d are real. In each cycle, the sum of the angles is
a submultiple of Stt : so that n + 1 conditions are thus imposed.
Again, the edges in a conjugate pair must be congruent; so that
n furthei' conditions are thus imposed. Accordingly, there remain
2ji — 4 real independent constants to determine the polygon.
The polygon thus dofcermined defines a Fuchsian function; as
the genus is zero, every function can be expressed rationally in
terms of x, so that the equation for v (leading to s, as the quotient
of two integrals) is
3 +lv=0,
ax'
where / is a rational function of x. Corresponding to the n + 1
cycles, there are w + 1 values of x ; let these be
Let OtTt be the sum of the internal angles of the zpolygon corre
sponding to (X,, so that Uj. is the reciprocal of an integer; and
take 0,1.^1 to be the quantity a for co . Then in the vicinity of a^,
we have
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161.] FUCHSIAN EQUATIONS
for each of the values of r. Thus, if we wribe
and remember that I is a, rational function of x, we have
«'«''=["]:.„■
for r=l, ...,n. In order to satisfy the condition for « = !»,
6(x) must be of order 2« — 2, and
aw 1(1 <■'«)«■"+•...
The number of coefficients in G (x) is 2m — 1 ; but the coefficient
of the highest power is known, and there are n relations among
the rest, owing to the conditions at CTi, ..., a„; lience there remain
w — 2 coefficients independent of one another. Each of these is
complex in general, so that they are effectively equivalent to
2w — 4 real constants. Assuming that the quantities Oj, .,., ««
are known, it is to be expected that the 2« — 4 conditions for the
polygon determine these 2n — 4 real constants.
; and we may take Os — O,
In the simplest
casc, \
ve have n
,=
<h
= 1, so that
/ = i
a?
■i(^
I)'
The conditions for .
X = 'K,
give
f
+
Hi
".=) +
J(l<
) +
= i (!«.■).
where a,, Kj, eta are the reciprocals of integers; the quantity /
then is the invariant of the hypergeometric series.
162. As another illustration, which may be treated somewhat
differently, consider the equation
1/' = a; (1  ic) (1  Qx\
where c is a real constant less than unity ; and write
ac = 1,
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502 EXAMPLE OF A [162.
SO that a is a, real constant greater than unity. Here, the points
ic= 0, 1, tt, X are ea«h of them singular; and the value of a is ^
for each of them. Consequently,
, l(ii), Hii) , i(ii) ,^ B c ,
' „'' '* (piy +(«»,)■ + ». + ^ ! + ««•
and the conditions for « = co give
One constant in / is left undetermined by these conditions ; thus
^="+(J^ + (S?S)'^l)(»!«)'
say, where \ is the undetermined constant. U is possible to
determine \, so that a: is a Fuchsian function of z, where z is
the quotient of two solutions of the equation
 + /d = 0.
dx'
As regaids this Fuchsian function, its polygon may be obtained
simply aa follows. We take four points A, B, G, D in the 2plane
to be the homologues of 0, 1, a, x ; owing to the value of a, which
is ^ in each case, the internal angies of the polygon must each be
\ir. We make the edges AB, CD conjugate, and likewise the
edges BG, DA ; and then there is a single cycle, ADGB, the sum
of the angles in which is %Tr. With the former notation, we thus
have 2 = 1, « = 2 ; so that
2p = 2 + 1  1 = 2,
and therefore p = 1, as should be the case. Further, the sum of the
angles of a curvilinear triangle, entirely on one side of the real
axis, is less than tt, when the centres of the circular arcs he on
the real axis : so that, if our polygon be curvilinear, the sum of its
angles would be less than 27r (for it could be made up of two
triangles), whereas the sum is actually 2Tr. Hence the polygon
can only be a rectangle, and the Fuchsian functions are doubly
periodic. We therefore take
a: = sn' ^, y^snzcnz dn z,
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62.] FUCHSIAN EQUATION
s is manifestly permissible ; and then
dx '2.y 2cM(a;l)i(a.'(i.)i'
which leads to
''■■"I • L^ + (a, 1)' + (»■(.)'] ^x{x^\){„a)
21,
so that we have
X._J(„ + 1).
This value of X renders x (and so y) a Fuehsian funetioa of the
quotient of two solutions of the equation
As regards the integrals of this equation, the indicial equation
of« = Ois
/>(pi) + iV».
SO that p = ^, p = I Denoting by Vi and v^ the integrals that
belong to J and J respectively, we have
! + §:>■+...
= ^j;i .
i'iii+c>e'+...
= en" f,
after the earlier analysis.
Similarly, in the vicinity of « = 1, wo find integrals
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604
EXAMPLE OF A
and then, taking
,^^.
we find
Now
»l
cn=
t
=
(1

0
«)(f2)' + i(l2»)(l«)(f^)'
so that
i, = (ol)HI;K).
Hence
!:<')' e^)'
so that, as
where AB
BC.
 1, because
we have
Again, i]
nthe
vicinity of « = a, we find integrals
[162.
cr,(^o)ii + 2A;^j''^(^») + ...,
and then, takini
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162.] FUCHSIAS EQUATION
we find
Also
ic — cr = sn' c
c
= — dn^ f
^—{KKiKj + }S^—^—''hKKiKy^...,
so that
Proceeding as before, this leads to the relations
Lastly, for large values of x, we have
F,«ilA(l+o)i+...l,
r,^ll(l + o)i+...};
and then, taking
w.
we find
Now
6<l+a)f.'+....
1 1
«snr
=csL"(fiin
= c(riir7ic(r!'Jr')'(i+<»)4
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506 INTEGRALS EXPRESSED AS [162.
SO that
Proceeding as before, this leads to the relations
If, = ci (v^  iK\) \
Wy = o^v, ] ■
The relations, in fact, have enabled us to construct the expressions
for each fundamental system in terms of the first and, therefore by
inference, in terms of every other.
Ex. 1. Discuss in the same way the Fuchsian differential equation
1 Ai f J 1 _\ 1 1^
connected with the eqiiation
Ex. 2. Shew that, if
whei'e p denotes Weieratrass's elliptic function,
'•■ ■•'•Ls^^lJ^ + (J:^' * (i:^'J"*(i«,)(»«,r(i^"S ■
and discuss the significance of the integral relation in regard to its paeudo
automorphic character for the equation
Es:. 3. 4 f d e tal J hg x the splane is composed of two semi
circles, one upo a d am t i tl c real axis for values of s correaponding to
values of a equal to and 1 the other upon a similar diameter for values of
a: equal to 1 ■uid u (wl e e > 1) and of two straight lines drawn, through
points corresj ondii^ to and a, perpendicular to the axis of real quantities.
Prove that tlie subsidiary equation of the second order, for the construction
of X as an automorphic function of the quotient of two of its integrals, is
id,' 'L"'^^ (»!)■ + (io)>J^^»(«l)(»o)'
where the constant jj is to be properly determined.
AUTOMORI'HIC BY'NCTIONS USED TO MAKE THE InTEGRALH
OF ANY Equation Uniform.
163. If, for any given equation, there is only one singularity,
it can be made to lie at the origin.
In order to obtain a variable s, in terms of which the integrals
of the given equation can be expressed uniformly, we construct an
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163.] UNIFORM FUNCTIONS .507
equation of the second order which has i« = for a singularity,
of such a form that the indicial ecjuation for x = has equal
roots (§ 160). This auxiliary equation may have other singulari
ties, but otherwise it may be kept as simple as possible. Such
an equation is
the indicial ecjuation for »: = is
e{dl)=X.
90 that \~ — I if it has equal roots. Thus the equation is
Two integrals are given by
Vi = x^, fla = 3^ log « ;
thU3
which is the new independent variable.
An equation of the kind indicated ia {§ 45, Ex. 6)
when the variable ia changed from :>: to s, where j; = e', the equation hccomea
The integrals are synectic for all finite values of 3.
164, When a given differential equation has two singularities,
a homographic transformation can be applied so as to fix them at
X = 0. iK=l.
To obtain a variable z in terms of which the integrals of the
given equation can be expressed uniformly, we construct an
equation of the second order, having and 1 as its singularities
and such that the respective indicial equations have repeated
roots. An appropriate equation is
d'v _ a + ^w
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508 EXPRESSION OF INTEGKALS [164.
The indicial equation for « = is
p{pl) = a,
so that a= — ^ ; the indicial equation f or ic = 1 is
p(pl) = «+/3.
SO that a + ^ = — ^, and therefore ,8 = 0, so that the equation is
One integral is easily found to be
and then z, the quotient of another integral by v,, is given by
dz ^C ^  1
dx jii' x(a!~l)'
on particularising the constant C, which may be arbitrary. Thus
gives a new variable g, Huch that the integrals oi' the given
differential equation arc uniform functions of s.
Thus let the equatio:
>a2' = <'.
which has :f=0 and x=l for real singularitiea ; it is easy to verify that
a:=ro ia not a Hingularity but only an ordinary point for every integral.
When the equation is transformed so that b is the indepeodent variable, it
becomes
the integrals of which clearly are uniform functions of s.
165. When a given differential equation has three singulari
ties, a homograpfiic transformation can be used so as to fix them
at a;=0, 1, oc .
We may proceed in two ways. It may be possible to choose,
as the fundamental region in the splanc, a triangle, having
circular arcs for its sides, and having Xtt, firr, vtt for its internal
angles at points which are the homologues of 0, oo , 1 respectively :
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165.] AS UNIFORM FUNCTIONS
X, fi,, V being the reciprocals of integers. Then the
equation may be taken in the form
i(^% iO^A^) iOr''') , i^'''^"''
which is the norma! form of the equation of the hypergeometric
series with parameters a, jS, 7, where
X' = a7)<, ^■ = (a«', „■ = (, = /3).
The variable s may be taken as the quotient of two solutions of
the subsidiary equation ; and so
.fCa + lT, ff + 1
")
It is known* that x, thus defined, is a uniform automorphic
function of s.
This transformation will render uniform the integrals of a
differential equation, which has no aiogularities except at 0, 1,
00 , provided the integrals are regular in the vicinity of those
singularities and belong to indices which are integer multiples of
X, V, fi respectively. If these conditions are not satisfied, in
particular, if the singularities are essential for the integrals, then
we proceed by an alternative methoi^.
We take a subsidiary equation having 0, 1, co for singularities,
such that the indicial equation for each of them has equal roots.
Let it be
where a', ^3", y are to be chosen so that the indicial ecjuation for
each of the singularities has equal roots. These equations are
p(pl) = a', <T(al) = a' + ^' + y', t(t + 1) = 7',
so that
"'i, f>' = i. 7'i,
and thus the equation is
• r. F., § 275.
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510 GENERAL APPLICATION OF [165.
The coefficient of v is the invariant of a hyp orgeo metric equation,
of which the parameters are
a = 0^l, 7 = 1;
so that s, the quotient of two integrals v, is also the quotient of
two integrals of the equation
''diu
i» = 0.
This is the equation of the quarterperiods in elliptic functions:
BO that
This relation effectively defines a; as a modular function* of z :
the fundamental region is a curvilinear triangle. The function
exists over the whole splane : the axis of real quantities is a line
of essential singularity.
Any differential equation, having a; = 0, 1, v> for all its singu
larities no matter what their character may he, can be transformed
by the preceding relation so that a is the independent variable ;
its integrals are then expressible as functions of z which are
uniform over the whole of the eplane, their essential singularities
lying on the axis of real quantities.
Ea:. A differential equation haa only tliree singularities at x=a, 6, e,
such that the roots of the indicial equations of those points are int<^er
multipleB of a, 8, y respectively, where a, ft y are reciprocals of integers.
Shew that a variable, in terms of which the integrals can bo expressed as
uniform functions, ia given by taking the quotient of two Riemann Pfunctioiis
with the appropriate singularities aiid indices.
AuTOMORPHic Functions applied to General Linear
Equations of any Order.
166. At the beginning of the preceding explanations and
discussions, it was assumed {§ 157) that all the singular values
of X are real. The assumption was then made for the sake of
simplicity : it can be proved"! to be unnecessary.
* T. F., % 303.
+ Poincare, Ada Math., I. iv, pp. 246—250.
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1.66.] AUTOMORPHIC FUNCTIONS 511
Firstly, let the singularities be constituted by a,, a^, ..., Om,
all of which are real, and by c, which will be supposed complex.
With these we shall associate Co, the conjugate of c ; and we write
4>{x) = {xc){a:c„),
a quadratic polynomial with real coefficients. Then all the
quantities
are real. Construct a fundamental region in the ^plane, such
that the foregoing m + 2 quantities are the homologues of the
comere ; and let
be the relation that gives the conformal representation of the
region upon half the Xplane, so that F(s) is a Fuchsian faoction
of 2.
Consider the variable x, as defined by the equation
So long as s remains within the fundamental region, a: is a
uniform function of e; it could cease to be so, only if
that is, if a! = ^c + ^c„, and then we should have
y(») = .f(ic + }o.),
which is not possible for values of z within the region. Also,
J is not zero for any value of s within the region; for then
we should have
which would make a zero magnification between the Xplane
and the ^region: this we know to be impossible for internal
^points. This uniform function x, whose derivative does not
vanish within the polygon, cannot acqiiirc either of the values
c or Co within the polygon, for then we should havo
F{s) = 0.
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512 AUTOMOBPHIC FUNCTIONS AND [166.
which is possible only at a comer. Nor can it acquire any of
the values til, eta, ..., a™ for points within the ;r polygon : for
at any such value, we have
i*" («) = *(»).
which again is possible only at a corner.
Now since X = l'(3) is a relation that con formally represents
the half Xplane upon a ^polygon bounded by circular arcs (this
polygon being otherwise apt for the construction of autornorphic
functions), we have (§ 157)
where ^ (X) is a rational function of X. But for any variables
X and X, we have
and therefore, in the present case,
[z, «) = 2 (2a; ~ c  c„f f(af cz c^ + cc„) 
= 2^(^),
say, where '^V(x) is a rational function of ie. Hence s is the
quotient of two integrals of the equation
S + '^W'"
Now cc is known to be a uniform function of s ; it is therefore a
Fuchsian function of z. And we have proved that, for values of z
within the polygon, x cannot acquire any of the real values
(ti, Oa, ..., (tm or either of the complex values c, d, and, further,
that ^ does not vanish.
as
Secondly, to extend this result to the case, when x is not
to acquire any one of any number of complex values for irpoints
within the polygon, we adopt an inductive proof; we assume the
result to hold when there are q — 1 pairs of conjugate complex
values, and shall then prove it to hold when there are q pairs. It
has been proved to hold, (i), when there are no complex values
and, (ii), when there is a pair of conjugate complex values : it thus
will be proved to hold generally.
(2^
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166.]
LINEAR EQUATIONS IN GENERAL
513
Suppose, then, that the given icsingnlarities arc made up of a
number m of real values a,,a.i, ...,fflm>and of a number of complex
values. Let the latter be increased in number by associating with
each complex value its conjugate complex, whenever that conjugate
does not occur in the aggregate ; and let the increased aggregate
be denoted by
arranged in conjugate pairs. Write
which is a polynomial of (
equation
fiee 2q with real coefficients. The
dx
= 0,
of degree 1q~l with real coeOicients, certainly p
root ; its other roots, when not real, can be arranged in conjuj
pairs, the number of pairs not being greater than g — 1. Let its
roots be denoted by
h, h, ..., Vi.
an aggregate which contains not more than 5 — 1 conjugate pairs.
In the series of quantities
0; <f.(aO, ■, 0(««); 0(M. ■■■. <^(6^.);
there are certainly m + 2 real quantities ; and there are not more
than 5 — 1 conjugate pairs of complex quantities. According to
our hypothesis, a Fuchsian function G{z) can be constructed, such
that the foregoing m + 2 + 2 (5  1) quantities are the homologues
of the comers of an appropriate fundamental region, and (?' (s)
does not vanish within the region. Then, proceeding on the same
lines as in the simpler case, we consider a variable .v, defined by
the relation
*W = (?{«).
So long as s remains within the fundamental region, ic is a
uniform function of 2 ; it could cease to be so, only if
that is, it'a: = 6,, 6j or b;q,, and then we should have
e(«)*(M, 4,(b,), .... or 4,(b„^o,
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514 FL'CHSIAN EQUATIONS HATING [lfi6.
which is not possible for values of s within the region. Also,
J does not vanish for values of z within the region ; for otherwise
we should have
for such values, and this is known not to be the ease. Further, m,
being a uniform function of z whose derivative does not vanish for
values within tbe polygon, cannot acquire any of the values c, or
Cf, for r=\, ...,q, within the polygon; if it could, we should have
(i (as) = there, and then
i'Wo,
which is possible only at a corner. Nor can it acquire any of the
values a,, ..., Om for values of ^ within the polygon ; if it could, we
should have
F{z) = 4,{<h), (f){a,), ..., or ^(0>
which again is possible only at a comer.
Now since Y, = G(z), is an automorphic function, it follows*
that
which is equal to — (e, Y], also is an automorphic function.
Consider the upper half of the Fplane. So far as the equation
Y = G(z) is concerned, certain points on the upper side of the
axis of real quantities are exceptional, not more than g — 1 in
number ; these can be considered as excluded, and cuts drawn
from them to singular points on the real axis. We then can
regard this simplyconnected and I'esolved half plane as conformally
represented upon the polygon by the equation F = G (s) ; hencef
where 5(F) is a rational function of F. But
where </> (x) is a polynomial ; hence
1^, «lh Yli^y + ir,]
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166.] ASSIGNED SINGULARITIES 515
say, where 0(a;) is a rational function of x. Hence z is the
quotient of two integrals of the equation
Now a; is known to be a uniform function of z. It is therefore a
Fuchsian function of z, which acquires the particular assigned
values only at the corners of the fundamental region and nowhere
within the region ; its derivative does not vanish anywhere within
the region.
The statement is thus established.
167. The preceding explanations, outlines of proofs, and
analysis, will give an indication of the kind of result to be
obtained, and the kind of application to differential equations to
be made. It will be recognised that such proofs as have been
adduced are not entii'ely complete : thus, when a number of real
constants is to be determined by the same number of equations,
whether algebraical or transcendental, it would be necessary to
shew that the constants, if determined in the precise number, are
real. As, however, it was stated at the beginning of these sections
that only an introductory sketch of the theory would be given,
there will be no attempt to complete the preceding proofs : we
shall be content with referring the student, for the long and com
plicated processes needed to establish even the existence of certain
results without evaluating their exact form, to the classical memoirs
by Poincar^, and to the treatise by Fricke and Klein, which have
already been quoted*. It may be convenient to recount the
most important and central results of Poincar^'s investigations,
which have any application to the theory of linear differential
equations.
Let
be a linear equation of order g, having rational functions of x and
y for its coefficients, where y is defined in terms of a; by the
algebraic equation
' A memoi) by E. T. Wliittaker, "On the eonneiion of algebraic funotions
with automorphic functions," Fhil. Trami. (1899), pp. 1—32, may also be consulted.
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516 POINCAEfi'S THEOREMS [167.
tbis equation in w will be called the main equation. Let
da:'
H'.y)
be another equation, in which Six, y) is a rational function of w
and y ; it will be called the subsidiary equation, and its elements
are entifeiy at our disposal.
Let cc = o,^, 2/ = &^,bea singularity of the main equation. If all
the integrals are regular at this singularity, if they aie i'ree from
logarithms, and if they belong to exponents, which are commensur
able quantities {no two being equal), let fc~' {where k is an integer)
be a quantity such that the exponents are integer multiples of ^'.
We make a; = a^, y = b^,& singularity of the subsidiary equation.
In the case of the indicated hypothesis as feo the integrals of the
main equation, we make the difference of the two roots of the
indioial equation of the subsidiary equation equal to k~^. In every
other case, we make those two roots equal. This is to be effected
for each of the singularities of the main equation.
Thus the subsidiary equation is made to possess all the
singularities of the main equation. It may have other singulari
ties also ; for each of them, the difference of the two roots of the
corresponding indicial equation is made either zero or the re
ciprocal of an integer, at our own choice. By these conditions, the
coefficient 6(x,y) will be partly determinate: but a number of
i will remain undetermined.
The effect of these conditions is, by the analysis of § 160, to
make x and y uniform functions of e, where z is the quotient of
two linearly independent integrals of the subsidiary equation ;
and no further conditions for this purpose need be imposed upon
the parameters, which may therefore be used to secure other
properties of the uniform functions. The various forms of 6,
corresponding to the various determinations of the parameters,
determine a corresponding number of differential equations ; all
of these are said to belong to the same type, which thus is
characterised by the singularities and their indicial (
Poincar^ has proved a number of propositions connected with
e results that can be obtained by the appropriate assignment
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167.] ON AUTOMORPHIC FUNCTIONS 517
of values to these parameters. Of these, the most important
I. It is possible to assign a iinique set of values in such
a way as to secure that x and y are Fuchsian functions of z,
existing only within a fundamental circle.
II. It is possible to assign sets of values, unlimited in
number, in such a way in each case as to secure that x and
y are Kleinian functions of z, existing over only part of the
2plane.
III. It is possible to assign a unique set of values in such
a way as to secure that x and y are Fuchsian functions or
Kleinian functions of z, existing over the whole of the 3plane.
There are limiting cases when the Fuchsian function becomes
doubly periodic, or simply periodic, or rational.
PoiKCAR^'s Theorem that any Likbar Equation can be
INTEGRATED BY" MEANS OF FuCHSUN AND ZeTAFUCHSIAN
Functions.
168. Consider now the integrals of the main differential
equation, when they are expressed in tenns of the variable z.
We shall assume that x and y have been determined as Fuchsian
functions oi z, existing only within the fundamental circle.
Near an ordinary point x^, y„, any integral w is a holomorphic
function oix — x„; near such a point, ic is a holomorphic function
of 2 — 2o ; so that w, when expressed as a function of 2, is a holo
morphic function of z.
In the vicinity of a singularity (a, b), there are two cases to
consider. If all the exponents to which the integrals belong are
commensurable quantities, so that they are integer multiples of
some proper fraction &^ where k is an integer, and if the integrals
are free from logarithms, then every integral is of the form
«. = (x»)'S(^«).
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518 POINCAR^'S THEOREM AS TO THE [168.
where >5 is a holomorphic function of x — a,. As in § 160, we have
SO that
where T and li are holomorphic functions. Hence
w = {zcYG{z~c),
where the function G is holomorphic in the vicinity of c. Thus w
is a uniform function of 3 ; if /i is positive, then c is an ordinary
point ; if ^ is negative, it is a pole.
In all other cases, whether the integrals involve logarithms, or
the exponents to which they belong are not all commensurable, or
the singularity is one where some of the integrals, or even all the
integrals, are irregular, the roots of the indicial equation for the
subsidiary equation are equal. In consequence, the two circular
arcs of any polygon touch, and thus the angular point is on the
fundamental circle. As we consider the values of z within the
fundamental circle, the character of the integral, when expressed
as a function of z, does not arise for the point of the kind under
consideration.
It thus appears that, when z is restricted to lie within the
fundamental circle of the Fuchsian functions which are the repre
sentative expressions of x and y, any integral of the main equation
is a uniform function of z. When this uniform function has poles,
it can be represented in the form
where the zeros of G, (z) are the poles of the integral in unchanged
multiplicity, and both 6 {s) and Qi {z) are holomorphic functions
of z, within the fundamental circle. When the uniform function
representing the integral has no poles, it can be expressed in the
form
where the function H (z) is holomorphic everywhere within the
fundamental circle.
Hence we have Poincare's theorem* that the integrals of a
linear differential equation with algebraic coefficients can be ex
pressed as uniform functions of an appropriately chosen variable.
* Acta Math., t. IV, p. 311.
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16!).] INTEGRALS OF LINEAR EQUATIOKS 519
169, The characteristic property of these uniform functions
can be obtaiued as follows. Takiug the equation in the form
where it is supposed that the term (if any) which involved ".^^i
has been removed from the equation by the usual substitution
(§ 152), we denote by 0,, 0^, .... 8,, a fundamental system of
integrals in the vicinity of any singularity (a^, i„). Let a closed
path on the Eiemann surface, associated with the permanent
equation, be described round the singularity; then, when the
path is completed, the members of the fundamental system
have acquired values ^i'. 6^, ..,, 6q, such that
e^ = a["^l e, + al'^le^ + ... + a^^^l e^. {« = i, 2, . . . , 7),
where the coefficients a*^' are constants such that their determ
inant is unity, because the derivative of order next to the highest
is absent from the differential equation.
Now ce and y are Fuchsian functions of z, existing only within
the fundamental circle in the zplane ; hence, when the path on
the Riemann surface, which cannot be made evanescent, is com
pleted, ic and y return to their initial values, and z has described
some path which is not evanescent. It follows, from the nature
of the functions, that the end of the spath is a point in another
polygon, homologous with the initial position, so that the final
position of z is of the form
a^e + Q ^
7.^ + K ■
The integrals dt, d„ ..., 6q are uniform functions of z\ let them
be denoted by 4>i{z), <fh(^)i •■■, 4'q(^) Moreover, B„' is the value
of Sa «t tJie conclusion of the path ; thus
"Wz + bJ'
so that the integrals in the fundamental system consist of a set of
uniform functions of z, which are characterised by the property
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520 ZETAFUCHSIA?J [169.
Corresponding to the substitution of the Fuehsian group, we have
a linear substitution S^ in the quantities 0i, 0^1 ■■. 'f'g the
aggregate of these linear substitutious 8^, forms a group, which
is isomorphic with the Fuchsian group.
Functions of this pseudoautomorphic character are called*
Zetafuchsian by Poincar^ : and thus we can say that linear differ
ential equations can be integrated by means of Fuchsian and Zeta
^cksian functions which are uniform. It is, however, necessary
to obtain explicit expressions for the functions 0, in order that
the equation may be regarded as integrated, This is effected
(I.e.) by Poiocar^ as follows.
Let
represent the substitution inverse to yS^, so that the quantities
A^^ are the minors of the determinant of S^. Take any q
arbitrary rational functions of z, say Hi{s), S^i^), .., Hqip); and
by means of them, in association with the Fuchsian group, con
struct p infinite series, defined by the equations
?,(.) = siA'''^ir.(
s + hii'^i^+^if""
for the q values 1, ...,qoi fi.; the quantity m is a positive integer;
and the summation with regard to i is over all the substitutions
of the Fuchsian group. This integer ni is at our disposal : by
choosing it sufficiently large, and by limiting the rational func
tions H, so that no one of the quantities
is infinite on the fundamental circle, all the series can bo made
absolutely converging: but we do not stay to establish this
resulff. Assuming this convergence, and writing
* Acta Math., t. v, p. 237.
t It can be establishea on the same liaes as the convergence of Poincai^'s
Thetafuchsian series: T. F., gg 304, 305.
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169.] FUNCTIONS 521
so that, for any value of k and al! the vahies of i, we get all the
values of p for the group, we have
But
Owing to the properties of the isomorphic groups, we have
and therefore
SkS,'' = sr.
that is,
and therefore
V7j;3 + 6i/ n=I l^"
Now let 0(2) represent a Thetafuchsian series*, with the
parametric integer m, and possessing the foregoing Kuchsian
group: then, for each substitution of the group, we have
^(S^')^(«^+«~^w
We introduce functions Zi, Z^, ..., Z^, defined by the relations
(f = l ?)■
They satisfy the conditions
and therefore we may take
or the q functions Z, which are Zetafuchaian functions, constitute
a system of integrals of the differential equation.
170. As regards the Zetafuchsian functions thus constructed,
it will be noted that the rational functions Hu ,.., Hq, which
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522 PR0PBBTIK8 OF A [170.
enter into their construction, are arbitrary; so that an infinite
number of Zetafuchsian functions can be formed, admitting a
Fucbsian group 6 and the linear group (say G) isomorphic with G.
Further, the Thetafuchsiaa series @(z) with the parametric
integer m is any whatever; but, as
we have
f^ _ f V \f' l'^^l±_§A __ ^
so that we may take
where P (cc, y) is any uniform function of le and y. The simplest
case occurs when P{ie,y) = \.
Again, we have
z C"^ "^
A\
"ZM')*
««2,
W+...K
'*'/.<^)i
"1?*^ +
sj"
and therefore
1
dZ.
<!§■
so that
t. IV + /3A V7,.
iT^:)=«^
dZ,
■0 i
d^S
■■C^
i
that is.
zj"^
z + 13.
:)=cf
+ "Z
{41 rf^,
da; \.7(
,z + S,
v. A, •
are a Zetafuchsian system, admitting the Fuchsian group G and
the isomorphic lineai group G.
The same property is possessed for all the derivatives of any
order of the system Zi, Z,^, ..., Z^ with regard to x.
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170,] ZETAFUGHSIAN SYSTEM 523
Tliis property is used by Poincare to obtain the most general
expression for a Zetafiichsian system, admitting the groups G and
Q. Let it be T^, T,, ..., T^; and construct the matrix
, dZ,
dt'Z,
T,
^" *.■■
■' d^^ '
, dZ,
d''Z,
T,
d''Z,
■' d^' '
T,
Denote by (— l)°'~'A,i_] the determinant obtained by cutting out
the a'" column from the matrix : then, by a known property of
determinants, we have
AA + A.! + ,.. + A,_
S^.''.
ues of ». Hence
„ a. . A, iZ,
■'"^"a, " A, ■&
A,_, di^Z^
When s is subjected to any transformation of .the group 0,
the quantities in any column in the matrix are subjected to the
corresponding linear transformation of the group G ; so that each
of the 5 + 1 determinants Ao, Aj, .... A^ is multiplied by the
determinant of the linear transformation. Hence A, h A^ is un
altered, that is, it is automorphic for the substitution of the
group G\ and therefore, as this property is possessed for each
substitution. A, ; A^ is automorphic for the group Q. Conse
quently, Ar ! A, is a rational function of x and y, say
?=^..
(fO, 1, . ..,?!)!
T,..FA + F,
dx
. + n
for 11 = 1, 2, ..., q. This is Poincar^'s expression for the most
general Zetafuchsian system, admitting the Fuchsian group G
and the isomorphic linear group G.
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524 CONCLUDING [170.
Wo can immediately verify that Z^, ..., Zg satisfy a linear
differential equation, having coefficients that are rational in x
and y. For
diZ d^Z^ d'iZ^
da^ ' dx'i'' '"' da:^ '
are a Zetafuchsian system, admitting the Fuchsian group G and
the isomorphic linear group G ; and therefore rational functions
t^o, 1^, ..., i^j_i exist, such that
holding for ail values of n. Thus Z^, ..., Zq are integrals of the
linear differential equation
Similarly, T„ ..., Tq are integra!s of a linear differential equation
also of order q, having rational functions of x and y for its co
efficients, and characterised by the same groups G and G as
characterise the equation satisfied hy Zi, ...,Zq.
Concluding Remarks.
171, The Zetafuchsian and Thetafuchsian functions thus
used occur, for the most part, in the form of series of a particular
kind; as they vpere first devised by Poincar^, his name is fre
quently associated with them. The main aim in constructing
them was to obtain functions which should exhibit, simply and
clearly, the organic character of automorphism under the substi
tutions of the groups; and they are avowedly intended* to be
distinct in nature from series adapted to numerical calculation,
such as series in powers of z.
Unless both these properties, viz. the exhibition of the organic
chai'acter of the function and its adaptability to numerical calcu
lation, are possessed by the functions involved, it is manifest that
they are not in the most useful form. It is unlikely that the
best development of the general theory can be effected, until
" Acta Math., t. v, p. 211.
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171.] REMARKS 625
functions have been obtained in a form that possesses both the
properties indicateci. In this connection, Klein* quotes a parallel
instance from the theory of elliptic functions, viz. the series of the
form
S S (mw + wi'ft)')"",
usedf by Eisenstein, which exhibit the characteristic antomorphic
property of the modular functions, but are not adapted to nu
merical calculation Their deficiency in this respect has been
met by the po&seBsion of the thetafunctions and the sigma
functions The generalisation of the Jacobian th etafunction
and the Weierstrassian sigma function, required for automorphic
functionb, has not yet been attained.
We thus letum to the statement made at the beginning of
the foiegoing sketch of Poincar^'s theory of linear differential
equations with algebraic coefficients. The explicit analysis con
nected with the theory of automorphic functions has not yet
acquired sufficiently comprehensive forms upon which to work ;
and therefore its application to linear differential equations, as to
any other subject, can be only partial and imperfect in its present
st^;e. The theory of automorphic functions in general presents
great possibilities of research : the gradual realisation of these
possibilities will be followed by corresponding developments in
many regions of analysis.
• Vorlesungen S. lineare Differentialgleiehungen d. sweiten Ordnting, (Gottingen,
1894}, p. 496. See also Fricke imd Klein, Theorie der automoTpken Functionen,
t. II, p. 156.
+ For references, see T.F.,% 56.
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INDEX TO PART III.
I. er
ed
g
2'i
'
M
ed
bl
54
t tl
257.
gul
Algebraic coeMeieDts, ei^uatiotis having,
Chapter x; oharnoter of integrals of,
in vioinity of branchpoint, 479, and
in vicinity of a singularity, 480 ; mode
of constructing integrals of, 483 ;
Appell's class of, 484 ; aseocisited with
automorphio fnnetiona, 488 (see auto
morphic fmictioas).
Algebraio equation, roots of, satisfy a
linear equation with rational ooefii
oienta, 46, 174 ; connected with differ
ential resolvents, 49.
Algebraic integrals, eq.uations having,
45, 165, Chapter v ; oonneoted with
theory of finite groups, 175 ; connected
with theory of oovariants, 175; equa
tions of second order having, 176 et
Beq. ; equations of thin] order having,
191 et seq. ; eqaationa of fourth order
having, 201; constraction of, 184,
198 ; and homogeneons forms, 202.
Aji^ytieid form of group of integrals
associated with multiple root of funda
mental equation of a singularity, 66 ;
likewise for multiple root of funda
mental equation for a period or periods,
416, 454.
Anuulus, integral converging in any (see
fundamental equation, irregular inte
gral, Lawent seriet).
tmormaJes, 270.
Apparent singularity, 117; conilitLons
for, 119.
Appell, 209, 484.
tl I 117
A t m pb t t d d ft t al
equationshavingalgebiaic coefficients,
48S ; and confonnal representation,
491 ; associated with linear equations
of second order, 495 ; constructed for
a speoid case, 600 ; when there is one
singularity, 6iD6; when there are two
singularities, 508 ; when there are
three, 509, 610; in general, 510 et
Barnes, 448.
begleitender bilinearer Differentialaua
dritek, 254.
Benoit, 474.
Bessel'E equation, 1, 13, 84, 100, 101,
126, 164, 330, 333, 393.
Bilinear ooDcomitant of two reciprocally
adjoint equations, 254.
Bdoher, 161, 169.
B6tlier's theorem on equations of Fuchs
ian type with five singularities, 161.
Boole, 229.
Boulanger, 195, 197, 198.
BrioscM, 206, a08, 218.
Casorati, 55, 60, 417.
Caochy, 11, 30.
Canohy'a theorem used to establish
existence of syneotic integral of a
linear eq nation, 11.
Cayley, 94, 113, 121, 182, 216, 233, 246,
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528
INDEX TO PAET 111
Cels, 2S4.
CharaoleriBtiG equation belonging to a
singalurity, 40.
Churacteristic equation toe deteimming
fitctor of normal integrals, 294 ; effect
of simple root of, 294; effect o£
multiple root, 398.
Cbaracterietic function of an equation,
Cbaracteristic index, definerl, 221 ;
and namlier of regulav integrals
of an equation, 930, 233 ;
of reciprooally adjoint equatione,
the eame, 257.
ChrjBtal, 7, 83.
Circular cjlinder, differential equation
of, 164 ; (see BeseeVi equation).
Class, equations of Fuchsian (see Ji'tic/is
ian type).
Cockle, 49.
CoefHcients, form of, near a eingularit;
if all integtalB there are regular, 78.
Collet, 20.
Oonformal representation, and auto
morphic functions, 491 ; ami funda
mental polygon, 493.
Constant ooefBoients, equation having,
14—20.
Construction, of regular integrals, liy
method of Frobeniua, 78; of normal
integrals, periodio integrals (see normal
integrals, simplyperiodie iniegraU,
doublypenodic integraU).
Continnation prooeae applied to ftyneetio
integral, 20.
Continaed fractions used to obtain a
fundamental equation, 439.
Covariants associated viitli algebraic
integrals, 202 ; for equations of third
order, 203, 209; for equations of
fourth order, 204 ; for equations of
second order, 206.
Craig, vi, 411.
Crawford, 474.
Curve, integral, defined, 303, HOS.
Darbous, 20, 254, 475.
Definite integrals (see Laplace's definite
integral, doubleloop integral).
Determinant of a system of integrals,
25 ; its value, 37 ;
not vanishing, the system is fun
damental , 29 ;
of a fundamentaJ system does not
vanish, 30 ;
il form of, for one particular
syst
, 34;
form of, near a. aingalarity, 77 ;
when the ooeffioienta are peri
odic, 406, 446 ; when the co
efB.cients are algebraic, 481.
Determinants, infinite (see infinite de
tertainanta).
Determining factor, of normal integral,
262; obtained by Thome's meHiod,
262 et seq.; conditions for, 265; tor
' itegrals of Hamburger's equations,
Diagonal of infinite determinant, 349.
Differencerelations, 63, 417.
Diffeiential invariants (see invariants,
differential).
Differential resolvents, 49.
Dini, 254, 256.
Divisors, elementary (see elementary
Doubleloop integrals,
integi'ate equations, mi ei seq.
Doublyperiodic ooefBoienta, eqnaiiona
having, 441 et seq. ; substitutions for
the periods, 413 ; ^ndamental equa
tions for the periods, 444, 445.
Doublyperiodio integrals of second kind,
447 ; Ficaid's theorem on, 447 ; num
ber of, 448, 450 ; belonging to Lamp's
equation, 463; how constructed, 471,
475.
applied t
Blement of fundamental system. 30.
Elementary divisors, of certain determ
inants, 41 — 43 ; of the fundamental
equation, 65; determine groups and
subgroups of integrals. 62 :
effect of, upon number of periodio
integrals when coefficients are
periodio, 416, 460.
Elliott, M„ 434, 425, 474.
Elliptic cylinder, differential equation
of, 164, 399, 431—441.
Expansion of converging infinite de
terminants, 363.
Expansions, asymptotic (see asymptotic
expamioiiK).
Exponent, to which regular integral
belongs, 74; properties of, 75;
to which the det«rminant of a
fundamental system belongs,
77;
to which normal integral belongs,
262;
of irregular integral as zero of an
infinite determinant, 368,
Exponents, snm of, for equations of
Fuchsian type, 128 ;
for Bieraann'a Pfnnotion, 139.
Fabry, 94, 270.
Factor, determining (see determining
Jaetoi).
Fano, 214, 218.
Finite groups of lineolinear substitu
tions, in one variable, 176; connected
with polyhedral functions, 181 ; asso
ciated with equations of second order
having algebraic int^rals, 182 ; used
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INDEX TO PART III
529
for con struct ion of algebra
185;
Integra
J vftriablea, 193 ; their dif
ferential invariants, 195; the
Laguerre invariaot, 196 ; used
to construct equations of third
order having algebraic integrals,
197;
in three variables, 300.
First kind, periodic function of, 410.
Floqnet, 231. 334, 259, 411, 418.
Frioke, 489. 515, 535.
Frobenius, 78. 93, 109, 336, 231, 233,
338, 247, 254, 257, 369.
Frobffliius' metliod, for the construction
of integrals (ail being regular), 79 et
aeq. ; variation of, suggested bj Cajley
for some cases, 114; applied to hyper
geometi^c equation for special cases,
147; for the construction of integrals,
when only some are regolar, 335 et
seq.; used for construction of irregular
integrals, 379 et seq.
Fuchs, L., 10, 11. 60, 65, 66, 79, 93, 94,
109, 110, 117, 133, 135. 126, 129, 15S,
306, 208, 216, 399, 462.
Fnchsian equations, 123, 495 et seq. ; in
dependent variable a uniform function
of quotient of integrals of. 496—199 ;
mode of determining coefficients in.
501; used as subsidiary to linearequa
tions of any order, 515.
Fnohsion functions, associated with
linear equations of the second order,
500, 502, 516 ; associated with linear
equations of general order. 615, 617;
in the expression of integrals as uni
form functions, 620.
Fnchsian group, {see Fuchsiaii /miction,
Zetafuchsiam fanetioii).
Focbsian type, equations of. Chapter iv,
pp. 133 et seq.; form of, 123; proper
ties of exponents, 126 ;
when fully determined by singu
larities and exponents, 128;
of second order with any number
of singularities, 150;
forms of, when fc is an ordinary
point, 152; when as is a singu
larity, 155, 158; Klein's normal,
158;
Lamp's equation transformed so
as to be of, 160;
equations of. haviag live singu
larities, 161; Bdcher's theorem
Fundamental equation, belonging to a
singularity, is same for all fuuda
mental systems, 38—40; invariants
of, 10; Poincar^'s theorem on, 40;
properties of, connected with ele
mentary divisors, 41—43;
fundamental system of integrals
assnoiated with 50 whe i roots
are simple 52 vhen a root is
multiple 53
loots of how lelated to roots of
mdioial equation 94
Fundamental equation when integrals
are iriegular expressed as an infinite
deteimmant i(89
luite terms 392
I methods of obtaining 399
Fundamental equations for double pen
ods 441 145 their form 447 loots
of determmedoubly peiioiio integrals
of the second Mud, 448 ; number of
these integrals, 150 ; efCeot of multiple
roots of, 451.
Fundamental equation for simple period,
406; is invariantive, 406; form of,
407 ; integral associated with a simple
root, 408; integrals associated with a
multiple root, 108; analytical expres
sion of, 419.
Fundamental equation when coefficients
are algebraic, 492; relation to in dicial
equation. 482.
Fundamental polygon for automorphic
functions, 490, 493, 500.
Fnndamental system of in tegrals, defined.
30; its deteiininant is not evanescent,
30; properties of , 30, 31 ; tests for,31,
32; form of, near singularity, 50; if
root of fundamental equation is sim
ple, 62; if root is multiple, 53;
affected by elementary divisors
of fundamental equation, 67;
aggregate of groups associated
with roots of indicia! equation
make fundamental system. 95.
Fundamental system, of irregular inte
grals, 387; of integrals when coelfioi
ents are simplyperiodic, 408, 419;
when ooeflioients are doublyjwriodio,
449 — 457; when coefGeients are alge
braic, 180.
Fundamental system, constituted by
group of integrals belonging to a
multiple root of fundamental equa
tion (see grou,^ of integraU).
Gordan, 183.
Grade of normal integral, 269.
Graf, 333.
GreenhUl, 466.
Glroup of tg 1 as led with mul
tiple t f f 1 tal equation,
53 ; r sol d t b g oups, by ele
mentary d 57 Hamborger's
subg p f 62 g al analytical
form t 5 1 fundamental
systen tit f 1 'er order, 72.
34
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530
INDEX TO PAET III
Group of iotegralB, associated with mul
tiple root of iudioial eqaation in methocl
of Frobeaiae, 80; general theorem on,
93; aggregate of , inal^e a fundamentSil
aystem, 96 ; compared with Ham
burger's groups, 113.
Group of integrals for hjpergeometrio
equation, 144.
Group of irregular integrals associated
with multiple root of characteristic
infinite determinant. 381 et seq.; re
solved into subgroups, 382.
Group of integrals associated with mul
tiple roots of fundamental equationsfoi'
periods when ooefficlents are doubly
periodic, 451 ; analytical expression of,
482, 467; fnriiier development of, when
uniform, 459.
Group of integrals associated with mul
tiple root of fundamental equation for
period when coefGcients are simply
periodic, 415 ; arranged in subgroups,
according to elementary divisors, 416;
analytioi expression of, 419; they
constitute a fondamental system for
equation of lower order, 420 ; further
expression of, when uniform, 421.
Groups of substitutions, finite (see^mi(e
groups);
infinite (see automorphic ftinc
Griinfeba, 359.
Gubler, 333.
Gtinthev, 11, 299, 399.
GjldSu, 469.
Halphen, 254, 356, 281, 316, 316, 448,
464, 465, 473.
Hamburger, 38, 60, 62, 63, 64, 113, 977,
380, 383, 286, 399, 489.
Hamburger's equations, 276 et seq. ;
of second order with normal ini
grals, 979; the number of nc
mai integrals, 280;
ot general order n witii normal
subnormal integrals, 288 et seq
of third order with normal or sod
normal integrals, 301 et seq.
Hamburger's subgroups of integrals (seo
subgroups of integrals j.
Hankel, 103, 333.
Harley, 49.
Heffter, 56, 156.
Heine, 164, 166, 431, 441.
Hermits, 15, 30, 448, 463, 465, 468, 473.
Hermite, on equation with constant
ooefUcienta, 15 — 20 ; on equation with
doublyperiodic ooeffioients, 465.
50.
Hiil, G. W., 348, 38
Hobson, 334, 337.
Homogeneous forms {i
398, 399, 403, 482.
linear equations, defined,
3; discussion limited to, 3.
Homogeneous relations between inte
grals when they are algebraic, 203,
217; of second degree for equations of
third order, 230; and of higher degree,
214.
Horn, 333, 341, 342, 346, 347.
Humbert, 167.
Hypergeometrio function, used to render
integrals of differential equations uni
form in speoial ease, 509.
Hypergeometrio aeries, equation of, 1, 13,
34, 103. 136, 144—160, 173, 338, 601,
509.
Identical relations, polynomial in powers
of a logarithm, cannot exist, 69.
Index, characteristic (see efiaracleiUUe
index) ; to which regular integral be
longs, 74; properties of, 75.
Indiojal eqaation, when all integrals are
regular, 85, 94; significance of, in the
method of Frobeuius, 85 ;
integral associated with a simple
root of, 86;
group of integrals associated with
a multiple root of, 86;
roots ot, how connected with roots
of fundamental equation, 94;
for equ t on w th not all integrals
reg a 223 2''7
ludioial f n t
regular J4
wh n not al ntegrals are regular,
227,
degree of, as affecting the number
of regular intf^Is, 230, 933 ;
of adjoint equation, as affecting
the number of regular integrals,
959.
Infinite determinant, giving exponent of
irregular integral, 368; modified to
another determinant, 369; is a peri
odic function of its parameter, 375 ;
effect of simple root of, 380. of a mul
tiple root of, 381 et seq. ; leads to the
fundamental equation of the singu
larity, 389 ; expressed in finite terms,
392.
Infinite determinants in general, 349;
convergence of, 360 ; properties of con
verging, in general, 353 et seq.; uni
form convergence of, wben functions
of a parameter, 358 ; may be capa
ble of differentiation, 359; used to
solve an unlimited number of linear
equations, 360 ; applied to construct
irregular integrals of differential equa
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INDEX TO PART III
531
Initial oondJtiouB defined, 4
values, 4 ; efCeol of, apon form of
synectic integral, 9
Integral curve, 203, 20S,
lot^rals. irregnkr (see i g la ate
grali).
Integrals, doublyperiodic irregular
normal, regular, simplype o!o b
norma], ayneotio (see unde these t ties
respeetivdy).
Intep^ rendered uuifona f net ons of
a variable, wben there s one singu
tarily, oOG ; when there are tno s Dgu
laritias, 608 ; when there are th ee
509, SIO; in general, 510 t sei by
1 of Z tat oh i n f ct ona 518
520
I
195
f
I 1 'f h
quat f
i
f tl thi d
fth f th
01 213
Laguerre'e, 196.
Invariants of fundamental equation, oon
neoted with singnlarity, 38, 40;
for irregular integrals, 398;
connected with a period or periods
405 i45
h fli t Ig b
il bl
p th
q t
th d f
Lab ange 251
I ague e 196
Lan^ eiuaton 1 12b 151 IbO 16o
168 338 148 4b4— 473
IJam^ a generahsed equation IbO
Laplace a defin te ntegral aat sf Tng
equat an vith ational ooefhc enta
318 oonto r of 23 developed nto
'.B here liie e exist
324 e
eq
isp eas ng an reg lar
ntegral 364 proof of convergence
w thin BJ1 annnlne 3b6
I/egeudre s equation 1 13 34 lOi 1 6
IbO 163
Liapounofl, 319, 42S — 431.
Liapounoft'a theorem, applied to evaluate
Laplace's definite integral, 324; me
thod of discussing uniform periodic
integrals, 425.
Lindemann, 4iU, 434, 437.
Lindstedt, 439.
Linear algebraic equations, infinite
Eyetem of, solved by meana of infinite
determinants, 360.
Linear difierential equs^tion, definition
of, a.
Lineolinear substitutiona (aee finite
L g
, quantity affected by, can
uniform linear differential
t and determine ita funda
tal atem, 66;
d tical relatione, polynomial in
fe lac iniegj^s free from, 106 j
CO dition that some regular
egral shall be free from,
tai ed by g 1 sat t F b
m th 1 379 th n.t t t Id
M d 11 333.
t 1 J tem 387
M k ft 19.
bl bym'^ ft
M f nfinite determinants. 354.
ph f t (ae in. pi
M tt g Leffler, 399, 463.
f )
M d 1 f notion, naed to reader inte
8 1 f differential eqaationa uniform
da 197 200 Sii 334 33 341
p al case, 610; Eiaenatain'a
g e 113
f aimUac to, 525.
M It 1 1 oot, group of integrala asso
I in 150 153 15 15 161 17b 1 S
tel th (see mi.itiyie root).
1S7 190 117 tb 4 9 515 2
M Itipli f periodic integral of second
1 m I f rm f eq t f
ki d 410; ia ft root of the funda
d d dF h typ 158
m tal quation of the period, 406.
m thod f eq ti f sec d
M th 4
d h m Ig b tefe 1
176
N m 1 f m, (after Frobenius) of equa
341
t h ing some inlegrala regular.
K h d4 35J 398 3')9 4 3
27 1 component factors of such
mm 14b
eq tion, and of a composite
( Iter Kieia) of equation of Fuohs
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